Holomorphic Functions of Exponential Type and Duality for Stein Groups with Algebraic Connected Component of Identity

HOLOMORPHIC FUNCTIONS OF EXPONENTIAL TYPE AND DUALITY F OR STEIN GR OUPS WITH ALGEBRAIC CONNECTED C OMPONENT OF IDENTITY S. S. Akbaro v Octob er 29, 2018 2 In tro duction Since in 193 0s L. S. Pontry agin published his famo us duality theorem for Abelia n lo cally compact gr oups [27], the following problem eng ages the imag ination of sp ecialists in harmonic analysis fro m time to time: is it p ossibl e to gener alize Pontryagin duality to non-Ab elian lo c al ly c omp act gr oups in such a way that the d ual obje ct has the same natur e as the initial one? As is known, the ﬁrs t attempts to generalize Pon tryagin duality did not meet this requirement: in the M. G. Krein theor y , for instance, the dua l ob ject b G for a group G is a blo ck-algebra [2 0] (but no t a group, unlike Pon tryagin theory ). Appar en tly , a deep p eculiarity in human ps yc hology manifests itself here, but such a harmless trait like asymmetry b et ween G and b G in the theory of representations – a t ra it that can b e compared with diﬀerence b etw een the left and the right in anatomy – leads to numerous and, bec ause of the changing with time under standing of what the notion of gro up should mean, contin uing attempts to build a duality theory , where, on the o ne hand, “ all” gr oups ar e cov ered, and, on the o ther, the Pon tryagin s ymmetry b etw een initial ob jects and their duals is preser v ed. In the category theor y language, the unique o ne ﬁtting fo r such sp eculative as pirations, one can formulate this task co rrectly using the following tw o deﬁnitions . 1. Let us ca ll a contra v ariant functor A 7→ A ∗ : K → K on a given category K a duality functor in K , if its squa re, i.e. the cov ariant functor A 7→ ( A ∗ ) ∗ : K → K , is isomo rphic to the identit y functor id K : K → K . K ∗   ? ? ? ? K ∗ ? ?     id K / / K (A) The passage to the dual group G 7→ G • in P ontry agin theory is an example of dualit y functor: the na tural isomorphism b et ween G •• and G here is the mapping i G : G → G ••    i G ( x )( χ ) = χ ( x ) , x ∈ G, χ ∈ G • On the contrary , s a y , in the categ ory of Banach space s the pa ssage to the dual Bana c h space X 7→ X ∗ is not a duality functor (b ecause there a re non-r eﬂexiv e B anach s paces). 2. Suppo se we hav e: (a) three categories K , L , M with t wo full and faithful cov ariant functors A : K → L and B : L → M deﬁning a ch ain of embeddings : K ⊂ L ⊂ M , (b) tw o duality functors K 7→ K • : K → K and M 7→ M ∗ : M → M such that the functors K 7→ B ( A ( K • )) and K 7→  B ( A ( K ))  ∗ are isomor phic: M ∗ / / M L B O O L B O O K A O O • / / K A O O (B) W e shall call this constructio n a gener alization of the duality • fr om the c ate gory K to the c ate gory L . On adding these terms in to o ur armour y , we ca n formulate the task we ar e discussing as follows: ar e ther e any gener alizatio ns of Pontryagin duality fr om the c ate gory of Ab elian lo c al ly c omp act gr oups t o the 3 c ate gory of arbitr ary lo c al ly c omp act gr ou ps, and if yes, then which namely? The catego ry diagra m (B) in this formulation b ecomes as follows: M ∗ / / M lo cally co mpact g roups O O lo cally co mpact g roups O O Abel ian lo cally co mpact groups O O • / / Abel ian lo cally co mpact g roups O O (C) – a nd the key example here is the dualit y theory for ﬁnite groups, which can b e re garded as a generalization of Pon tryagin duality from the catego ry of Ab elian ﬁnite gr oups to the catego ry of all ﬁnite g roups: ﬁnite d i mensional Hopf algebras H 7→ H ∗ / / ﬁnite d i mensional Hopf algebras ﬁnite groups C G 7→ G O O ﬁnite groups C G 7→ G O O Abel ian ﬁnite groups O O G 7→ G • / / Abel ian ﬁnite groups O O (D) (here G 7→ C G is the pa ssage to g roup alg ebra, and H 7→ H ∗ the passage to dual Hopf algebr a). This example, apart fro m everything else, illustra tes another guiding idea of the “genera l dua lit y theo ry”: if we wan t to reduce the representations o f gr oups to repres en tations o f a lgebras, a nd ther efore cla im tha t the categ ory M consists of as socia tiv e alge bras, then these a lgebras H m ust hav e some supplemen tary structure, whic h allows us to endow dual ob jects H ∗ with a natur al structure of asso ciative algebra. Natural ob jects of that kind in general algebr a a re Hopf algebras. As a cor ollary , the constructions in “duality theor y” usually resemble Hopf algebr as, altho ugh a s a rule diﬀer from them, except for triv ial cases when, for instanc e, the alg ebra has ﬁnite dimensio n. A g eneralization of Pon tryagin duality as it is pre sen ted in diagra m (C ) was suggested in 1 973 in- depe nden tly by L. I. V ainerman and G. I. Kac from one side ([43], [44], [45]), and by M. Eno ck and J.-M. Sch wartz fro m another ([11], [1 2], [13]). The theory of Ka c a lgebras they develop ed summarize d a series of attempts made by diﬀerent mathema ticians, histo ry of which, as well a s the theory o f Ka c algebras itself, one can learn from m onog raph [10] b y Enock and Sch w artz. After 1973 the w ork in t his di- rection did not ceas e, because on the one hand, some improv ements were added int o the theory (again see details in [10]), and on the other , after the discov ery o f q uan tum groups in 1980s , mathematicians b egan to gener alize t he Pon tryagin duality to this class as well. Moreov er, the latter work is not ﬁnished by now – the theor y of lo cally compa ct quantum groups which app eared on this wav e is b eing actively dev elop ed by J. Kustermans , W. Pusz, P . M. So ltan, S. V aes, L. V ainerman, A. V an Daele, S. L. W oronowicz (an impression of this topic one ca n ﬁnd in collective monogr aph [28]). The idea o f multip lier Hopf algebra suggested by A. V an Daele in 19 90-s [41, 42], see ms to b e stra tegic in these inv estigations. Despite the active work and impressive enth us iasm demonstrated by mathematicia ns engaged in this theme, the theories they sug gest have a se rious shortcoming : al l the envelopi ng c ate gories M in these the ories c onsist of obje cts, which ar e formal ly not H opf algebr as. The Kac algebras, for instance, although 4 being c hosen a s a sub class among the o b jects called Hopf-von Neumann algebra s, in fact ar e no t Hopf algebras , because from the p oin t of view of ca tegory theor y in the deﬁnition of Hopf-v on Neuma nn algebras , unlike the pure algebr aic situation, tw o diﬀeren t tensor pro ducts are used simu ltaneous ly – the pro jective tensor pro duct for multiplication, and the tens or pro duct of v on Neumann algebras for comultiplication. In the theory of lo cally compact qua n tum gr oups the situation on this p oint seems to give even less hop es, b ecause here the c laim that the co m ultiplication must indisp ensably act into tensor pro duct is r ejected: in accorda nce with the ab o ve men tioned V an Daele’s idea of multiplier Hopf alg ebras the comultiplication here is deﬁned as an op erator from A into the algebr a M ( A ⊗ A ) o f multipliers on tensor pro duct A ⊗ A (which is chosen here as minimal tenso r pro duct of C ∗ -algebra s, see [21, 2 8]). In this pap er we suggest another a pproach to the generalization of Pontry agin duality , wher e this unpleasant eﬀect do es not o ccur: in our theory the en veloping categ ory M consists of “true” Hopf algebras , of course in the ca tegorical sense, i.e. the Hopf algebra s deﬁned in the same way as the usua l Hopf algebr as, but a fter replacing the catego ry of vector spac es by a given symmetrical (in the more general case – bra ided) monoida l catego ry (such Ho pf a lgebras a re sometimes called Hopf monoids, se e [29, 35, 36, 49]). The mission we set to o urselves is not quite a genera lization of P ontryagin duality to the class of lo cally compac t gro ups, as it is presented in diag ram (C), but a so lution of the same problem in the class of complex Lie groups. The fact is that a mong the four main branches of mathematics , where the idea of inv ariant int egr ation manifests itself – – general top ology (where the gr oups with inv ariant integral ar e exactly lo cally compact gro ups), – diﬀerential geometr y (where this r ole b elongs to Lie gro ups), – complex analysis (where the reductive co mplex groups can be r egarded as the groups with in tegra l), and – algebra ic geometry (again with reductive c omplex gr oups), – at least in the three ﬁrst disc iplines the generaliza tion of Pon tryagin dualit y has sense. In top ology this problem takes form of diagr am (C), in diﬀeren tial geometry one can consider the problem of g eneralization of Pon tryagin duality from Abelia n compactly generated Lie gro ups, say , to all compactly genera ted Lie groups, and in the complex a nalysis one can consider the pr oblem of genera lization of Pon tryagin dualit y from the class of Abelian compactly generated Stein g roups to the class of reductive co mplex groups (following [46], by a reductive gro up we mean complexiﬁca tion of a compact rea l Lie gro up). W e giv e a solution for the third of these problems. W e begin our considerations with the algebra O ( G ) of holomo rphic functions o n a complex Lie g roup G , since this is a natural functional algebra in complex analysis (in the o ther three disciplines this r ole b elongs to the alg ebra C ( G ) of contin uous functions, the algebra E ( G ) of smo oth functions, and the a lgebra R ( G ) of po lynomials). The idea we suggest as a heuristic hypothesis in this pap er – and its justiﬁcatio n we see in our work – is as fo llo ws. It seems likely , that to each of the three ﬁr st disciplines in our list – general top ology , diﬀerential geometry and complex analysis – from the p oint of view of the top ologica l alge bras used in these disciplines , corr esponds so me class of seminor ms, intrinsically co nnected to this class of algebra s. What those seminorms should b e in general top ology and in diﬀere n tial geometry we cannot say right now, but the class of submult iplicative seminorms, i.e. seminorms deﬁned by the ineq ualit y p ( x · y ) 6 p ( x ) · p ( y ) , is intrinsically connected with complex ana lysis (this idea is inspired to us by A. Y u. P irko vskii’s results on the Arens- Mic hael envelopes o f to polog ical algebr as – see [26]). T o g iv e exact meaning to the words “intrinsically connec ted”, we need to int ro duce the following tw o functors and one categ ory of Hopf alge bras: 1) The Ar ens-Michael en velop e A 7→ A ♥ . The a lgebras whose top ology is genera ted b y submultiplica- tive seminorms and satisﬁes supplementary condition of completeness, are called Arens- Mic hael 5 algebras (we discus s them in § 5). Each top olog ical algebra A has a nearest “fro m outside” Arens- Michael a lgebra – we denote it by A ♥ – called Arens-Michael env elop e of A . This alg ebra is a completion of A with res pect to the sy stem of a ll contin uous s ubm ultiplicative seminor ms on it. 2) Passage to dual ster e otyp e Hopf alg ebr a H 7→ H ⋆ . In the work [1] w e discussed in detail the symmetrical monoidal categ ories ( Ste , ⊙ ) and ( Ste , ⊛ ) of stereo t yp e spaces (with injective ⊙ a nd pro jective ⊛ tensor products). The Hopf algebra s in these categories are called respec tiv ely injective and pro jective stereotype Hopf alg ebras. The passa ge to the dual stereotype space H 7→ H ⋆ establishes an antiequiv alence b et ween these ca tegories o f Hopf alge bras. 3) Cate gory of holomorphic al ly r eﬂexive Hopf algebr as. The functors ♥ and ⋆ a llo w us to consider the category of pro jective and a t the same time injective stereo t yp e Hopf algebr as H , for which the successive application of the op erations ♥ and ⋆ alwa ys lea ds to injective and at the same time pro jective Hopf algebra s a nd, if we b egin with ♥ , then at the fourth step this chain leads ba c k to the initial Hopf alg ebra (of course up to an iso morphism). W e depict this clos ed chain by the following r eﬂexivity diagr am , H  ♥ / / H ♥ _ ⋆   H ⋆ _ ⋆ O O ( H ♥ ) ⋆  ♥ o o (E) and call such Hopf alge bras H holomorphic al ly r eﬂexive (see accurate deﬁnition in § 5(e)). The duality functor in the c ategory of such Hopf algebras is, certainly , the op eration H 7→ ( H ♥ ) ⋆ . The main res ult of our work, which clariﬁes our hints a bout “se minorms, intrinsically connected with complex a nalysis”, is that the algebr a O ⋆ ( G ) of a nalytical functionals on a compactly g enerated Stein group G with a lgebraic connected comp onent of iden tity is a holomor phically r eﬂexiv e Ho pf algebr a. T o prov e this we in tro duce in this pa per the algebra O exp ( G ) of holo morphic functions o f exp onent ial type on gr oup G , which is a subalgebra in the algebr a O ( G ) o f all holo morphic functions on G . The reﬂexiv it y diagram (E) for O ⋆ ( G ) is as follows: O ⋆ ( G )  ♥ / / O ⋆ exp ( G ) _ ⋆   O ( G ) _ ⋆ O O O exp ( G )  ♥ o o (F) F or the case of Ab elian groups the op eration ♥ b ecomes naturally isomorphic to the usual F ourier 6 transform, so this dia gram takes the for m: O ⋆ ( G )  ♥ F ourie r transform / / O ( G • ) _ ⋆   O ( G ) _ ⋆ O O O ⋆ ( G • )  ♥ F ourie r transform o o (G) (the dual group G • for a n Abelia n compactly generated Stein g roup G is deﬁned a s the group o f all homomorphisms from G into the multiplicative group C × := C \ { 0 } of no n-zero complex num b ers). This, in par ticular, implies the iso morphism of functor s O ⋆ ( G • ) ∼ =   O ⋆ ( G )  ♥  ⋆ , which gives a generaliza tion of Pontry agin duality in the complex case, and the diagr am (B) here takes the form: holomorphically reﬂexive Hopf algebras H 7→ ( H ♥ ) ⋆ / / holomorphically reﬂexive Hopf algebras compactly g enerated Stein groups with algebraic comp onent of identi t y O ⋆ ( G ) 7→ G O O compactly g enerated Stein groups with algebraic comp onent of identity O ⋆ ( G ) 7→ G O O Abel ian co m p actly gene rated Stein groups O O G 7→ G • / / Abel ian co m p actly gene rated Stein groups O O (H) Since every reductive gr oup is alg ebraic, this indeed will b e a solution of the third pro blem in our list. W e show in addition that the holomo rphic dua lit y we introduce here do es not limit itself to the class of compactly generated Stein gro ups with algebraic connected comp onen t of identit y , but extends to quantum g roups. As an exa mple we consider the qua n tum group ‘ az + b ’ of quan tum aﬃne automorphisms of complex pla ne (see [48, 40, 4 7, 28]). W e pr o ve that ‘ a z + b ’ is a holomorphica lly reﬂexive Hopf alg ebra in the sense of o ur deﬁnition. The author thanks sincerely D. N. Akhieser, O . Y u. Aristov, P . Ga uc her, A. Y a. Helemskii, A. Huckle- ber ry , E. B. Katsov, Y u. N. Kuznetsov a, T. Maszczyk, S. Y u. Nemiro vskii, A. Y u. Pirkovskii, V. L. Popov, P . So ltan, A. V an Daele fo r innumerous consultations and he lp during the work on this pa per. Besides that the idea o f pro of of Pro positio ns 4.1, 7 .12 and Lemma 7.9 b elongs to Y u. N. K uznetso v a. § 0 Stereot yp e spaces Stereotype spa ces we a re sp eaking a bout in this section were studied by the a uthor in detail in [1] (see also [2, 3]). § 0. STERE OTYPE SP ACES 7 (a) Deﬁnition and t ypical examples Let X be a lo cally conv ex space ov er C . Deno te b y X ⋆ the s pace o f all linear contin uous functionals f : X → C , endow ed with the top ology o f uniform conv ergenc e on totally b ounded sets in X . The space X is called ster e otyp e , if the natura l mapping i X : X → ( X ⋆ ) ⋆ | i X ( x )( f ) = f ( x ) , x ∈ X , f ∈ X ⋆ is an isomor phism of lo cally conv ex spa ces. Clearly the fo llo wing theorem holds: Theorem 0.1. If X is a s t er e otyp e s p ac e, then X ⋆ is also a ster e otyp e sp ac e. It turns out that ster eot yp e spa ces fo rm a very wide class, what ca n b e illustrated by the following diagram: ✬ ✫ ✩ ✪ STEREOTYPE SP ACES ✬ ✫ ✩ ✪ quasicomplete barreled spaces ✬ ✫ ✩ ✪ F r´ echet spaces ✗ ✖ ✔ ✕ Banach spaces ✬ ✫ ✩ ✪ reﬂexive spaces F r´ echet spa ces a nd Ba nac h space s will b e of sp ecial interest for us in this picture, s o we sha ll consider them in detail. Example 0 .1 ( F r ´ ec het spaces and Brauner spaces). Ev ery F r´ echet space X is stere ot yp e [7]. Its dual spa ce Y = X ⋆ is also ster eot yp e by theorem 0 .1. If { U n } is a countable lo cal ba se in X , then the po lars K n = U ◦ n form a countable fundamental system of c omp act sets in Y : every compact se t T ⊆ Y is contained in some co mpact set K n (this means by the wa y that Y canno t b e F r´ echet space, if X inﬁnite dimensional). The space s Y dual to F r´ echet spaces X (in the sense o f o ur deﬁnition) were originally considered by K. Brauner in [7 ], and we call them Br auner sp ac es . Their characteris tic prop erties ar e listed in the following prop osition. Prop osition 0.1. F or a lo c al ly c onvex sp ac e Y the fol lowing c onditions ar e e quivalent: (i) Y is a Br au n er sp ac e; (ii) Y is a c omplete Kel ley sp ac e (i.e. every set M ⊆ Y that has a close d interse ction M ∩ K with any c omp act set K ⊆ Y , is close d in Y ) and has a c ountable fu n damental system of c omp act sets K n : for e ach c omp act set T ⊂ Y ther e exists n ∈ N 1 such that T ⊆ K n ; (iii) Y i s a ster e otyp e sp ac e and has a c ountable fundamental syst em of c omp act sets K n : for e ach c omp act set T ⊂ Y ther e ex ist s n ∈ N su ch t hat T ⊆ K n ; 1 Ev erywhere in our paper N means the set of non-negative intege rs: N := { 0 , 1 , 2 , 3 , ... } . 8 (iv) Y is a ster e otyp e sp ac e and has a c ountable ex hausting syst em of c omp act set s K n : S ∞ n =1 K n = Y . Pr o of. The countable fundamental s ystem o f compact sets in Y is the sy stem of p olars K n = ◦ U n of a lo cal ba se U n in the dual F r´ ec het spa ce Y ⋆ . Mo dulo this rema rk all the statements in pro positio n 0.1 are obvious, except one – that a Brauner space Y is always a Kelley spa ce. This result b elongs to K . Brauner and is deduced in his pap er [7] fr om the Ba nac h-Dieudonn´ e theor em (see [17]). Corollary 0.1. If Y is a Br aun er sp ac e with t he fundamental system of c omp act sets K n , then any line ar mapping ϕ : Y → Z int o a lo c al ly c onvex sp ac e Z is c ont inuous if and only if it is c ontinuous on e ach c omp act set K n . Example 0.2 ( Bana c h spaces and Smith spaces). These are sp ecial cases of F r´ echet spa ces and Brauner spaces . If X is a Bana c h space, then X and Y = X ⋆ are stereotype spaces [31]. The p olar K = B ◦ of a ba ll B in Y is a universal c omp act set in Y , i.e. a co mpact set that swallows any other compact set T in Y . The spaces Y = X ⋆ dual to Banach spaces X (in the s ense of o ur deﬁnition) were originally considered by M. F. Smith in [31] – that is why w e ca ll them Smith s p ac es . Their character istic prop erties are listed in the following prop osition: Prop osition 0.2. F or a lo c al ly c onvex sp ac e Y the fol lowing c onditions ar e e quivalent: (i) Y is a Smith sp ac e; (ii) Y is a c omplete Kel ley sp ac e and has a u niversal c omp act set K : for any c omp act set T ⊂ X t her e exists λ ∈ C such t hat T ⊆ λK ; (iii) Y is a ster e otyp e sp ac e with a universal c omp act set; (iv) Y is a ster e otyp e sp ac e with a c omp act b arr el. Corollary 0. 2. If Y is a Sm ith sp ac e with a un iversal c omp act set K , t hen a line ar mapping ϕ : Y → Z into a lo c al ly c omp act sp ac e Z is c ontinuous if and only if it is c ontinuous on K . The connections b etw een the spaces of F r´ echet, Brauner, B anach and Smith ar e illustra ted in the following diagr am (where turnov er corresp onds to the passag e to the dual class ): ✬ ✫ ✩ ✪ F r ´ echet spaces ✬ ✫ Banach spaces ✬ ✫ ✩ ✪ Brauner spaces ✩ ✪ Smith spaces ﬁnite dimensional spaces (b) Smith space generated b y a compact set Let X be a stereotype space and K a n abso lutely conv ex compact set in X . W e de note by C K the linear subspace in X , g enerated by the set K : C K = [ λ> 0 λK (0.1) Endow C K by the K el ley top olo gy gener ate d by c omp act set s λK : a set M ⊆ C K is considered c losed in C K , if its intersection M ∩ λK with a n y co mpact s et λK is closed in X (or, equiv alen tly , in λK ). § 0. STERE OTYPE SP ACES 9 Theorem 0 .2. The Kel ley t op olo gy on C K , gener ate d by c omp act sets λK , is a unique top olo gy on C K , which turn s C K into a Smit h s p ac e with the universal c omp act set K . Pr o of. Denote by ( C K ) ′ the set of all line ar functiona ls on C K , contin uous o n the co mpact set K : f ∈ ( C K ) ′ ⇐ ⇒ f : C K → C & f | K ∈ C ( K ) Clearly , ( C K ) ′ is a Banach space with re spect to the no rm || f || = max x ∈ K | f ( x ) | (formally this turns ( C K ) ′ int o a clo sed s ubspace in C ( K )). Note tha t functionals f ∈ ( C K ) ′ separate po in ts in K , b ecause those of them who are restrictions on C K of functionals from g ∈ X ⋆ already posses s this prop erty: ∀ x, y ∈ K  x 6 = y = ⇒ ∃ g ∈ X ⋆ g ( x ) 6 = g ( y )  This means tha t the weak top ology σ on K , g enerated b y functionals f ∈ ( C K ) ′ , c oincides with the initial top ology τ of this compact set (b ecause σ is Hausdorﬀ and is ma jorized by τ ). This implies in its turn that the top ology of the spa ce ( C K ) ′ is the top ology of uniform co n vergence on ( C K ) ′ -weak absolutely co n vex compact sets of the form λK (these s ets for m a saturated s ystem in C K ). Hence by the Mack ey-Arens theore m [32], the sys tem (( C K ) ′ ) ⋆ of linear contin uous functionals o n ( C K ) ′ coincides with C K : C K = (( C K ) ′ ) ⋆ W e see that C K can b e identiﬁed with the space of linea r c on tinuous functionals on ( C K ) ′ . As a corolla ry , C K can b e endowed with the top olog y of dual space (in stereotype sens e) to the Banach spa ce ( C K ) ′ : C K ∼ = (( C K ) ′ ) ⋆ (0.2) This topolo gy turns C K into a Smith space, and, b y Prop osition 0.2, it coincides with the Kelley topo logy on C K , gener ated by compa ct sets λK . Let us denote this top ology on C K b y κ , and s ho w that it is a unique topo logy under which C K is a Smith space with the universal compact set K . Indeed, if ρ is ano ther top ology on C K with the same prop erty , then the identit y mapping ( C K ) ρ → ( C K ) κ is contin uous on K (since it preserves the top ology on K ), hence by Corolla ry 0.2, it is a con tinuous mapping of Smith spa ces. In the same wa y , the inv erse mapping ( C K ) κ → ( C K ) ρ is contin uous, a nd this means that the top ologies κ a nd ρ coincide. Corollary 0.3. The t op olo gy of the sp ac e C K c an b e e quivalently describ e d as the t op olo gy of un ifo rm c onver genc e on se quenc es of functionals { f k } ⊂ ( C K ) ′ tending to zer o, i.e. as the top olo gy gener ate d by seminorms of the form: p { f k } ( x ) = sup k ∈ N | f n ( x ) | wher e f k is a se quenc e of line ar functionals on C K , c ontinuous on K , and such t hat max t ∈ K | f k ( t ) | − → k →∞ 0 . Prop osition 0.3. F or any two absolutely c onvex c omp act sets K, L ⊆ X such that K ⊆ L , the natur al imb e dding of the Smith sp ac es which they gener ate ι L K : C K → C L is a c ontinuous mapping. Pr o of. The mapping ι L K is contin uous o n the compact set K , hence, by Cor ollary 0 .2, it is contin uous o n all C K . 10 (c) Brauner spaces generated b y an expanding sequence of compact sets A sequence of abso lutely convex co mpact sets K n in a stereotype space X will b e called exp anding , if ∀ n ∈ N K n + K n ⊆ K n +1 F o r every such sequence the set ∞ [ n =1 C K n = [ n ∈ N ,λ ∈ C λK n = [ n ∈ N K n is a subspa ce in the vector space X . W e endow it with the Kel ley top olo gy gener ate d by c omp act sets K n : a s et M ⊆ S ∞ n =1 K n is co nsidered closed in S ∞ n =1 K n , if its intersection M ∩ K n with any compac t set K n is closed in X (and equiv alently , in K n ). The following pr opo sition is pr o ved similarly with Theo rem 0.2: Theorem 0 .3. The Kel ley top olo gy on S ∞ n =1 K n , gener ate d by c omp act sets K n , is a unique top olo gy on S ∞ n =1 K n , u nder which t his sp ac e is a Br aun er sp ac e with the fundamental se quenc e of c omp act sets { K n } . Corollary 0.4. The top olo gy of the sp ac e S ∞ n =1 K n c an b e e quivalently describ e d as t he top olo gy of uniform c onver genc e on se quenc es of functionals { f k } ⊂ ( S ∞ n =1 K n ) ′ tending to zer o, i.e. as the t op olo gy gener ate d by s eminorms of t he form: p { f k } ( x ) = sup k ∈ N | f k ( x ) | wher e f k is an arbitr ary se quenc e of line ar functionals on S ∞ n =1 K n , c ontinu ous on e ach K n , su ch that ∀ n max t ∈ K n | f k ( t ) | − → k →∞ 0 . (d) Pro jective Banac h systems and injectiv e Smith systems Let X b e a lo cally conv ex space. A s tandard construction in the theo ry of top ologica l vector spaces assigns to ea c h abs olutely co n vex neighborho o d o f zero U in X a B anach space, which it is conv enient to denote by X/U and to call a quotient sp ac e of X over the neighb orho o d of zer o U . It is deﬁned as follows. First, we consider the set Ker U = \ ε> 0 ε · U, which is calle d kernel of t he neighb orho o d of zer o U – this is a closed subspace in X , since U is absolutely conv ex. Then we construct the quotient space X/ Ker U , and endow it with the top ology of normed s pace with U + Ker U as unit ba ll (this top ology in general is weaker than the usual top ology of quotient spa ce on X / Ker U ). This no rmed space X/ Ker U is usually not complete. Its completion is declared the ﬁnal result: X/U := ( X/ Ker U ) H (0.3) (here H means co mpletion). Theorem 0. 4. L et X b e a ster e otyp e sp ac e. F or any absolutely c onvex c omp act set K ⊆ X the S mith sp ac e it gener ates, C K , is c onne cte d with t he qu otient sp ac e X ⋆ /K ◦ of t he dual sp ac e X ⋆ with r esp e ct to the neighb orho o d of zer o K ◦ thr ough the formula ( C K ) ⋆ ∼ = X ⋆ /K ◦ (0.4) § 0. STERE OTYPE SP ACES 11 Let X b e a stereotype space and let K b e a exp anding sy stem of absolutely conv ex compact s et in X , i.e. a system satisfying the following co ndition: ∀ K, L ∈ K ∃ M ∈ K K ∪ L ⊆ M By Pr opo sition 0.3, for any compact sets K, L ∈ K such that K ⊆ L , the Smith spaces they g enerate are connected through a natural linea r contin uous ma pping ι L K : C K → C L . Since, obviously , for any three compact sets K ⊆ L ⊆ M the corr esponding mappings ar e connected throug h the equality ι M L ◦ ι L K = ι M K , the arising system of mappings { ι L K ; K , L ∈ K : K ⊆ L } is a n injective s ystem in the categ ory Ste of stereotype s paces. Like any other injective system in Ste , it has a limit – this is the pseudo completion of its lo cally conv ex injectiv e limit [1, Theo rem 4.2 1]: Ste - lim − → K →∞ C K =   LCS - lim − → K →∞ C K   ▽ The dual constructio n is often used in the theory of top ological vector space s. Let X be a stereotype space and suppos e U is a de cr e asing sys tem of absolutely con vex neighborho o ds of zero in X , i.e. a system satisfying the following c ondition: ∀ U, V ∈ U ∃ W ∈ U W ⊆ U ∩ V F o r a n y tw o neig h b orho ods of zero U, V ∈ U such that V ⊆ U the Banach spaces they ge nerate X/ Ker U and X/ Ker V are connected with ea c h o ther through a natur al linear con tinuous ma pping π V U : X/ Ker V → X/ Ker U . If we consider thr ee neig h b orho ods of zero W ⊆ V ⊆ U the c orresp onding mappings are connected throug h the equality π V U ◦ π W V = π W U , This means that the sys tem o f ma ppings { π V U ; U , V ∈ U : V ⊆ U } is a pr o jective sy stem in the categor y Ste of ster eot yp e spa ces. Its limit is the pseudo saturation of its loc ally convex pro jective limit [1 , Theorem 4.21]: Ste -lim ← − 0 ← U X/U =   LCS -lim ← − 0 ← U X/U   △ F r om (0.4) we hav e: Theorem 0.5. If K is an exp anding system of absolutely c onvex c omp act set s in a st er e otyp e sp ac e X , then the system of p olars U = { K ◦ ; K ∈ K } is a de cr e asing system of absolutely c onvex neighb orho o ds of zer o in the dual sp ac e X ⋆ . The limits of these systems ar e dual t o e ach other:   Ste - lim − → K →∞ C K   ⋆ = Ste - lim ← − 0 ← U X ⋆ /K ◦ (0.5) Example 0.3. Let K = K ( X ) be a system of a ll absolutely convex compact sets in X . Then the limit of the injective system { ι L K ; K, L ∈ K : K ⊆ L } in the catego ry of stereo t yp e spaces is the saturation 2 of the space X : Ste - lim − → K →∞ C K = X N 2 Saturation X N of a lo cally conv ex space X was deﬁned in [1, 1.2]. 12 Example 0.4. Dually , if U = U ( X ) is the s et o f all absolutely conv ex neig h b orho ods of zero in X , then the limit o f the pr o jective s ystem { π V U ; U , V ∈ U : V ⊆ U } in the c ategory of stereotype spa ces is the completion of the space X : Ste -lim ← − 0 ← U X/U = X H Theorem 0. 6. If K n is an ex p anding se quenc e of absolutely c onvex c omp act sets in a st er e otyp e sp ac e X , then the limit of the inje ctive system C K n in the c ate gory of st er e otyp e sp ac es c oincides with lo c al ly c onvex limit of this system and with the Br auner sp ac e gener ate d by t he se quenc e K n : Ste - lim − → n →∞ C K n = LCS - lim − → n →∞ C K n = ∞ [ n =1 C K n (0.6) (e) Banac h represen tation of a Smith space If X is a Banach space (with the nor m || · || X ), then let us deno te by X ∗ its dual Banach sp ac e in the usual sense, i.e. the space o f linear contin uous functionals on X with the norm || f || X ∗ = sup || x || X 6 1 | f ( x ) | If ϕ : X → Y is a contin uous linea r mapping of Ba nac h spa ces, then the symbo l ϕ ∗ : Y ∗ → X ∗ denotes the dual mapping: ϕ ∗ ( f ) = f ◦ ϕ, f ∈ Y ∗ The natural mapping from X into X ∗∗ will b e denoted b y s X : s X : X → X ∗∗ . Let Y b e a Smith space with the universal co mpact set T . Denote by Y B the normed space with Y as supp ort and T as unit ball. Theorem 0.7. F or any Smith sp ac e Y Y B ∼ = ( Y ⋆ ) ∗ . (0.7) henc e, Y B is a (c omplete and so a) Banach s p ac e. Pr o of. Consider t he space X = Y ⋆ . Then Y b ecomes a space of linear con tin uous functionals on a Banach space X with the top ology of uniform conv ergence on compact sets in X . The universal compact T in Y bec omes a p olar of the unit ba ll B in X . If we endow Y with the top olog y , where T is a unit ball, this is the same as if we endow Y with the to polog y of the normed space, dual to X : Y B = X ∗ = ( Y ⋆ ) ∗ . Since the (Ba nac h) dual to a Banach space is alwa ys a Banach space, Y B m ust b e a Banach space as well. W e call the space Y B Banach re pr esentation of the Smith spa ce Y . Note that the natural mapping ι Y : Y B → Y , ι Y ( y ) = y is univ ersal in the following sense: for any Ba nac h space Z and for an y linear contin uous mapping ϕ : Z → Y ther e exists a unique linear contin uous mapping χ : Z → Y B such that the following diag ram is commutativ e: Y B Y Z / / ι Y _ _ ? ? ? χ ? ?      ϕ § 0. STERE OTYPE SP ACES 13 If ϕ : X → Y is a linear contin uous mapping of Smith spa ces, then it turns the universal compact set S in X into a compact set ϕ ( S ) in Y , which, like any other co mpact set, is contained in so me homothety of the universal compa ct set T in Y : ϕ ( S ) ⊆ λT , λ > 0 This means that the mapping ϕ , b eing considered as the mapping b et ween the corresp onding Banach representations X B → Y B , is also contin uous. W e shall denote this mapping by the symbol ϕ B ϕ B : X B → Y B and call it Banach r epr esen t ation of t he mapping ϕ . Obviously , ϕ B ∼ = ( ϕ ⋆ ) ∗ (0.8) and the following diagr am is commutative X Y X B Y B / / ϕ O O ι X / / ϕ B O O ι Y (f ) I njectiv e systems of Banac h spaces, generated b y compact sets The follo wing co nstruction is o ften us ed in the theory of topo logical vector spaces. If B is a b ounded absolutely conv ex c losed set in a lo cally co n vex space X , then X B denotes the space S λ> 0 λB , endow ed with the to polog y of normed s pace with the unit ba ll B . If X B turns out to be complete (i.e. a B anach) space, then the set B is ca lled a Banach disk . F rom theorem 0.7 we have Prop osition 0 .4. Every absolutely c onvex c omp act set K in a ster e otyp e sp ac e X is a Banach disk, and the Banach sp ac e X K gener ate d by K is t he Banach r epr esent ation of t he Smith sp ac e C K : X K = ( C K ) B If K is a n expanding system of absolutely conv ex compa ct s ets in a s tereotype space X , then, a s w e told ab ov e, K generates an injective system of Smith spaces { C K } K ∈K , ι L K : C K → C L ( K, L ∈ K , K ⊆ L ) . Spec ialists in top ologica l vector space s used to r eplace this system of Smith s paces with the sys tem of the corre sponding Banach represe n tations: { ( C K ) B } K ∈K , ( ι L K ) B : ( C K ) B → ( C L ) B ( K, L ∈ K , K ⊆ L ) The following result s ho ws that the limits o f thos e systems co incide in the case when the injections ι L K : C K → C L are c ompact mapping s. Theorem 0 .8. L et K b e an ex p anding system of absolutely c onvex c omp act sets in a ster e otyp e sp ac e X . Then the fol lowing c onditions ar e e quivalent: (i) for any c omp act set K ∈ K ther e is a c omp act set L ∈ K such t hat K ⊆ L and the mappi ng of Smith sp ac es ι L K : C K → C L is c omp act, (ii) for any c omp act set K ∈ K ther e is a c omp act set L ∈ K such t hat K ⊆ L and t he mapping of Banach s p ac es ( ι L K ) B : ( C K ) B → ( C L ) B is c omp act. 14 If these c onditions hold then the lo c al ly c onvex inje ctive limits of the systems { C K } K ∈K and { ( C K ) B } K ∈K c oincide, LCS - lim − → K →∞ C K = LCS - lim − → K →∞ ( C K ) B (0.9) and the same is tru e for their ster e otyp e inje ctive limits: Ste - lim − → K →∞ C K = Ste - lim − → K →∞ ( C K ) B (0.10) If in addition to (i)-(ii) t he system K is c ountable (or c ontains a c ountable c oﬁnal su bsystem), then t hose four limits c oincide, Ste - lim − → K →∞ C K = LCS - lim − → K →∞ C K = LCS - lim − → K →∞ ( C K ) B = Ste - lim − → K →∞ ( C K ) B (0.11) and deﬁne a Br auner sp ac e. W e shall premise the pro of of this theo rem by several auxiliary prop ositions. First we note that by c omp act mappi ng of ster eot yp e spaces we mean what is usually meant, i.e. a linear contin uous mapping ϕ : X → Y such tha t ϕ ( U ) ⊆ T for some neighborho o d of ze ro U ⊆ X and so me compact set T ⊆ Y . Prop osition 0.5. L et X and Y b e Smith sp ac es and ϕ : X → Y a line ar c ontinuous mappi ng. The fol lowing c onditions ar e e quivalent: (i) ϕ : X → Y is a c omp act mapping; (ii) ϕ ⋆ : Y ⋆ → X ⋆ is a c omp act mapping; (iii) ϕ B : X B → Y B is a c omp act mapping. Pr o of. The eq uiv alence (i) ⇔ (ii) is o b vious, and (ii) ⇔ (iii) follows from (0.8) and a classica l result on compact mappings [3 0, Theorem 4.19 ]. Prop osition 0 . 6. If ϕ : X → Y is a c omp act mapping of Banach sp ac es, t hen its ( Banach) se c ond dual mapping ϕ ∗∗ : X ∗∗ → Y ∗∗ turns X ∗∗ into Y : ϕ ∗∗ ( X ∗∗ ) ⊆ Y (0.12) Pr o of. Let B b e unit ball in X and T a co mpact set in Y s uc h that ϕ ( B ) ⊆ T By the bipola r theorem, B is X ∗ -weakly dense in the unit ball B ◦◦ of the space X ∗∗ . Hence fo r each z ∈ B ◦◦ one can choo se a net z i ∈ B tending to z X ∗ -weakly B ∋ z i X ∗ -w eakly − → i →∞ z ∈ B ◦◦ (0.13) Since ϕ , like any weakly compac t op erator, turns X ∗ -weak Cauch y nets in to ‘str ong’ Cauch y nets, w e obtain that ϕ ( z i ) is a Cauch y net in the compact set T , hence it tends to so me y ∈ T . ϕ ( z i ) Y − → i →∞ y § 0. STERE OTYPE SP ACES 15 F r om this w e have: ∀ g ∈ Y ∗ ϕ ∗∗ ( z i )( g ) = ϕ ∗ ( g )( z i ) = g ( ϕ ( z i )) − → i →∞ g ( y ) = i Y ( y )( g ) (0.14) (where i Y : Y → Y ∗∗ is a natural embedding). On the other hand from (0.1 3) we also o btain that ∀ f ∈ X ∗ f ( z i ) − → i →∞ f ( z ) and, in par ticular, this must b e true for functionals f = ϕ ∗ ( g ) = g ◦ ϕ , wher e g ∈ Y ∗ : ∀ g ∈ Y ∗ ϕ ∗∗ ( z i )( g ) = ϕ ∗ ( g )( z i ) − → i →∞ ϕ ∗ ( g )( z ) = ϕ ∗∗ ( z )( g ) (0.15) F r om (0.14) and (0.15) we hav e ϕ ∗∗ ( z ) = y I.e., ϕ ∗∗ ( z ) ∈ Y . This holds for each p oint z ∈ B ◦◦ , so (0.12) must b e true. Prop osition 0. 7. L et ϕ : X → Y b e a line ar c ontinuous mapping of ster e otyp e sp ac es. F or any absolutely c onvex close d set V ⊆ Y the fol lowing formula holds: ϕ − 1 ( V ) = ◦  ϕ ⋆ ( V ◦ )  (0.16) Pr o of. x ∈ ◦  ϕ ⋆ ( V ◦ )  ⇐ ⇒ ∀ g ∈ V ◦ | ϕ ⋆ ( g )( x ) | = | g ( ϕ ( x )) | 6 1 ⇐ ⇒ ⇐ ⇒ ϕ ( x ) ∈ ◦ ( V ◦ ) = V ⇐ ⇒ x ∈ ϕ − 1 ( V ) Prop osition 0.8. Le t σ : X → Y b e a c omp act mapping of Smith sp ac es, and V an absolutely c on- vex close d n eigh b orho o d of zer o in the Banach r epr esentation Y B of the sp ac e Y . Then its pr eimage ( σ B ) − 1 ( V ) = σ − 1 ( V ) is a neighb orho o d of zer o in t he sp ac e X (and not only in its Banach r epr esentation X B ). X B Y B X Y   ι X / / σ B   ι Y / / σ Pr o of. By fo rm ula (0 .8), w e can consider σ B as a mapping of dual Ba nac h s paces for X ⋆ and Y ⋆ : σ B = ( σ ⋆ ) ∗ : ( X ⋆ ) ∗ → ( Y ⋆ ) ∗ Let us consider the Ba nac h dual mapping ( σ B ) ∗ = ( σ ⋆ ) ∗∗ : ( Y ⋆ ) ∗∗ → ( X ⋆ ) ∗∗ . Since σ ⋆ is compact, from Prop osition 0.6 it follows that ( σ ⋆ ) ∗∗  ( Y ⋆ ) ∗∗  ⊆ X ⋆ This can b e illustrated by the dia gram ( X ⋆ ) ∗∗ ( Y ⋆ ) ∗∗ X ⋆ Y ⋆ o o ( σ ⋆ ) ∗∗ w w o o o o o o o O O i X o o σ ⋆ O O i Y 16 Now we hav e: V is a neighborho o d o f zero in Y B = ( Y ⋆ ) ∗ ⇓ V ◦ is a bo unded se t in ( Y ⋆ ) ∗∗ ⇓ ( σ ⋆ ) ∗∗ ( V ◦ ) is a totally b ounded set in ( X ⋆ ) ∗∗ (since ( σ ⋆ ) ∗∗ is a compact mapping) & ( σ ⋆ ) ∗∗ ( V ◦ ) ⊆ X ⋆ ⇓ ( σ ⋆ ) ∗∗ ( V ◦ ) is a totally b ounded set in X ⋆ ⇓ ( σ B ) − 1 ( V ) = (0.16) = ◦  ( σ B ) ∗ ( V ◦ )  = ◦  ( σ ⋆ ) ∗∗ ( V ◦ )  is a neighbor hoo d of zero in X Pr o of of The or em 0.8. The equiv alence of (i) and (ii) immediately fo llo ws from Prop ositions 0.5. After that formula (0.9) follows fr om P ropo sition 0.8: if U is an absolutely conv ex neigh b orho o d of zero in LCS - lim − → K →∞ ( C K ) B , i.e. U = { U K ; K ∈ K} is a system of neighbor hoo ds of zer oes in the spaces ( C K ) B , satisfying the condition ∀ K, L ∈ K  K ⊆ L = ⇒ U K = ( ι L K ) − 1 ( U L )  then from Pr opos ition 0.8 it follows that all those neighbo rho ods are neighborho o ds o f zero in the spa ces C K . This prov es the co n tinuit y of the mapping LCS - lim − → K →∞ C K → LCS - lim − → K →∞ ( C K ) B The contin uity of the inv erse mapping is obvious. Thus, the top ologie s in those spa ces coincide. In other words, those lim its coincide, and we obtain (0.9). This implies in its turn (0.10). Finally if K is countable, then by Theor em 0.6 the lo cally conv ex limits coincide with ster eot yp e limits, and we hav e (0.11) . (g) Nuclear stereot yp e spaces A mapping of stereot yp e spaces ϕ : X → Y w e shall call nucle ar , if it satisﬁes the following four equiv alent conditions: (i) there exist tw o sequences f n X ⋆ − → n →∞ 0 , b n Y − → n →∞ 0 , λ n > 0 , ∞ X n =1 λ n < ∞ , such that ϕ ( x ) = ∞ X n =1 λ n · f n ( x ) · b n , x ∈ X ; (0.17) (ii) there exist t w o tota lly bo unded s equences { f n } in X ⋆ and { b n } in Y , and a num ber sequence λ n > 0 , P ∞ n =1 λ n < ∞ such that (0.17) holds; § 0. STERE OTYPE SP ACES 17 (iii) there exist tw o bounded sequences { f n } in X ⋆ , and { b n } in Y , and a n umber sequence λ n > 0, P ∞ n =1 λ n < ∞ such that (0 .17) holds ; (iv) there exist a sequence of functionals { f n } ⊆ X ⋆ , equicontin uous on X , and a sequence of vectors { b n } in so me Ba nach disk B ⊂ Y , and a num ber seq uence λ n > 0 , P ∞ n =1 λ n < ∞ such that (0.1 7) holds. Clearly , a nuclear mapping is a lw ays co n tinu ous. Pr o of. The implicatio ns (i) ⇒ (ii) ⇒ (iii) are o b vious. Let us pr o ve the implication (iii) ⇒ (i). Suppo se (iii) is true , i.e. the se quences { f n } ⊂ X ⋆ and { b n } ⊂ Y in the formula (0 .17) are just b ounded. Let us choose a num ber s equences σ n > 0 such that σ n − → n →∞ + ∞ , ∞ X n =1 λ n · σ n < ∞ (w e can denote P ∞ n =1 λ n = C for this, and then divide the sequence λ n int o blo cks P n k 0, n 0 = 0, and put σ n = k + 1 for n k < n 6 n k +1 ). Then we can replace f n , b n , λ n with f f n := f n √ σ n , e b n := b n √ σ n , f λ n := λ n · σ n , bec ause ∞ X n =1 f λ n · f f n ( x ) · e b n = ∞ X n =1 λ n · σ n · f n ( x ) √ σ n · b n √ σ n = ∞ X n =1 λ n · f n ( x ) · b n = ϕ ( x ) And since f n and b n are b ounded, then f f n and e b n tend to zero f f n − → n →∞ 0 , e b n − → n →∞ 0 , ∞ X n =1 f λ n < ∞ i.e. (i) holds. It r emains now to show that the conditions (i)-(iii) are eq uiv alent to (iv), i.e . the standard deﬁnition of nuclear mapping [3 2, 17]. The implication (iv) ⇒ (iii) is obvious. On the other ha nd, the implication (ii) ⇒ (iv) holds: fr om (ii) it follows that the sequence of functionals { f n } , b eing tota lly b ounded in X ⋆ , is equicontin uous on X by [1, The orem 2.5], a nd the sequence { b n } b elong to Banach disk T = absconv { b n } by theorem 0.4. This is exa ctly (iv). Theorem 0 .9. A line ar c ontinuous mapping b etwe en ster e otyp e sp ac es ϕ : X → Y is nu cle ar if and only if its dual mapping ϕ ⋆ : Y ⋆ → X ⋆ is nucle ar. Let us c all a mapping o f stereo t yp e s paces ϕ : X → Y quasinucle ar , if for any compact set T ⊂ Y ⋆ there exist a s equence of functionals { f n } ⊆ X ⋆ equicontin uous o n X a nd a num b er s equence λ n > 0, P ∞ n =1 λ n < ∞ such that max g ∈ T | g  ϕ ( x )  | 6 ∞ X n =1 λ n | f n ( x ) | (0.18) If X and Y are Banach spaces , then quasinuclearity of ϕ : X → Y means inequality || ϕ ( x ) || Y 6 ∞ X n =1 λ n | f n ( x ) | , (0.19) where || · || Y is the norm in Y , f n a b ounded sequence o f functionals on X , λ n a summable num b er sequence. 18 Lemma 0.1. L et ϕ : X → Y b e a line ar c ont inuous mapping of Banach sp ac es, and let ϕ ∗ : Y ∗ → X ∗ b e its Banach dual mapping. Then – if ϕ is nu cle ar, then ϕ ∗ is nucle ar as wel l, – if ϕ ∗ is nucle ar, then ϕ is quasinucle ar. Pr o of. The ﬁrst pr opo sition her e is obvious (and w ell-known, see [25, 3.1.8 ]) a nd implies the sec ond one: if ϕ ∗ : Y ∗ → X ∗ is n uclear , then ϕ ∗∗ : X ∗∗ → Y ∗∗ is also nuclear, a nd as a co rollary , quasinuclear. F r om this we hav e that ϕ is also quas in uclear, since it is a restrictio n (and a corestriction) o f a quasinuclear mapping ϕ ∗∗ on the subspace X ⊆ X ∗∗ (and subspace Y ⊆ Y ∗∗ ). As usual, we call a stereotype space X nucle ar [25], if every its cont inuous linear ma pping into an arbitrar y Banach space X → Y is nuclear. Theorem 0.10 (Br auner, [7 ]) . A Br auner sp ac e X is nucle ar if and only if its dual F r´ ech et sp ac e X ⋆ is nucle ar. Theorem 0. 11. L et K b e an exp anding system of absolutely c onvex c omp act sets in a ster e otyp e s p ac e X . Then the fol lowing c onditions ar e e qu ivale nt: (i) for any c omp act set K ∈ K ther e is a c omp act set L ∈ K such t hat K ⊆ L and the mappi ng of Smith sp ac es ι L K : C K → C L is nucle ar, (ii) for any c omp act set K ∈ K ther e is a c omp act set L ∈ K such t hat K ⊆ L and t he mapping of Banach s p ac es ( ι L K ) B : ( C K ) B → ( C L ) B is nucle ar, If these c onditions hold, t hen the lo c al ly c onvex inje ctive limits of the systems { C K } K ∈K and { ( C K ) B } K ∈K c oincide, LCS - lim − → K →∞ C K = LCS - lim − → K →∞ ( C K ) B (0.20) and the same is tru e for their ster e otyp e inje ctive limits: Ste - lim − → K →∞ C K = St e - lim − → K →∞ ( C K ) B . (0.21) If in addition to (i)-(ii) t he system K is c ountable (or c ontains a c ountable c oﬁnal subsystem), then al l those four limits ar e e qual Ste - lim − → K →∞ C K = LCS - lim − → K →∞ C K = LCS - lim − → K →∞ ( C K ) B = Ste - lim − → K →∞ ( C K ) B (0.22) and deﬁne a nucle ar Br auner sp ac e. Pr o of. All statements here follow from Theor em 0.8, except the equiv alence b et ween (i) a nd (ii). (i)= ⇒ (ii). If ι L K is nuclear, then ( ι L K ) ⋆ is nuclear by Theor em 0.9. Hence ( ι L K ) B = (( ι L K ) ⋆ ) ∗ is nuclear by Lemma 0.1. (i)= ⇒ (ii). F or a given compac t set K let us choo se a compa ct se t L ⊇ K suc h that ( ι L K ) B is nuclear. Similarly , tak e a compact set M ⊇ L such that ( ι M L ) B is n uclear . Then f ro m nuclearity of ( ι L K ) B = (( ι L K ) ⋆ ) ∗ and ( ι M L ) B = (( ι M L ) ⋆ ) ∗ by Lemma 0 .1 we hav e that ( ι L K ) ⋆ and ( ι M L ) ⋆ are quasinuclear. fro m this we have that ( ι L K ) ⋆ ◦ ( ι M L ) ⋆ is nuclear as a comp osition o f quasinuclear mappings [25, 3.3.2]. Now by Theorem 0.9, ι M L ◦ ι L K = ι M K bec omes nuclear, as a stereo t yp e dual mapping. (h) Spaces C M and C M The spaces C M and C M we a re talking ab out here are usua lly mentioned in textbo oks on topo logical vector spaces as o b jects for exercise s ([5, Chapter IV, § 1, Exer cise 1 1,13], [32, Chapter IV, E xerscise 6]). W e list here some of their prop erties for the further r eferences. § 0. STERE OTYPE SP ACES 19 Space of functions C M . Let M b e an arbitr ary set. Denote by C M the lo cally convex space of a ll complex-v alued functions on M , u ∈ C M ⇐ ⇒ u : M → C with the p oint wise op erations, and the to polog y of po in twise co n vergence on M , i.e . the top olog y gener - ated by seminor ms | u | N = sup x ∈ N | u ( x ) | (0.23) where N is an arbitra ry ﬁnite set in M . W e call C M the sp ac e of functions o n the set M . Note that C M is isomor phic to the direct pro duct of card M copies of the ﬁeld C : C M ∼ = C car d M As a coro llary , C M is alwa ys nuclear (since nuclearity is inherited by direc t pro ducts [32 , Theor em 7.4]). Theorem 0.12. F or a lo c al ly c onvex sp ac e X over C the fol lowing c onditions ar e e quivalent: (a) X ∼ = C M for some set M ; (b) X is a c omplete sp ac e with the we ak t op olo gy 3 ; (c) X is a sp ac e of minimal typ e 4 . Theorem 0.13. C M is a F r´ echet sp ac e if and only if t he set M is at most c oun table. The space of p oin t charges C M . Let a gain M b e an arbitrary set. Denote by C M the s et of all nu mber families { α x ; x ∈ M } indexed by elements of M and satisfying the following ﬁniteness condition: all the num b ers α x , but a ﬁnite subfamily , v anish α ∈ C M ⇐ ⇒ α = { α x ; x ∈ M } , α x ∈ C , card { x ∈ M : α x 6 = 0 } < ∞ (0.24) (clearly the families { α x ; x ∈ M } can be c onsidered as functions α : M → C with ﬁnite support). The set C M is endow ed with p oint wise alg ebraic op erations (sum and mult iplication by a scalar ) and a to polog y generated by seminor ms | α | r = sup | u | 6 r |h u, α i| = X x ∈ G r ( x ) · | α x | (0.25) where r : G → R + is an arbitrar y no nnegative function o n M . W e call C M the s p ac e of p oint char ges on the set M (and its elements – p oint charges on M ). W e c an note that C M is isomor phic to a lo cally conv ex direct sum of card M copies of the ﬁeld C : C M ∼ = C car d M As a coro llary , C M is nuclear if and o nly if M is at mos t countable (nuclearity is inherited by c oun table direct sums [32, Theorem 7.4], and if M is not countable, then the imbedding C M → ℓ 1 ( M ) is not a nu clear ma pping). Theorem 0.14. F or a lo c al ly c onvex sp ac e X over the ﬁeld C the fol lowing c onditions ar e e quivalent: (a) X ∼ = C M for some set M ; (b) X is a c o c omplete 5 Mackey sp ac e, wher e every c omp act set is ﬁ nite-dimensional; 3 A lo cally con vex space X is called a sp ac e with the we ak top olo gy , if its top ology is generated by the seminorms of the form | x | f = | f ( x ) | , where f ar e l inear contin uous functionals on X . 4 A lo cally conv ex space X is called a sp ac e of minimal typ e , if there i s no weak er Hausdorﬀ lo cally conv ex topology on X . 5 W e say that a lo cally con ve x space X is c o-c omplete , if every linear functional f : X → C which is contin uous on ev ery compact set K ⊂ X , is contin uous on X . 20 (c) top olo gy of X is maximal in the class of lo c al ly c onvex top olo gies on X (i.e. the r e is no str onger lo c al ly c onvex top olo gy on X ). The following theorem in esse n tial part b elongs to S. Kakutani and V. Klee [18]: Theorem 0.15 . C M is a Br au n er sp ac e if and only if the set M is at most c ountable. In this (and only in this) c ase t he top olo gy of C M c oincides with the s o c al le d ﬁnite top ology on C M (a set A is said to b e close d in ﬁn ite top olo gy, if its interse ction A ∩ F with any ﬁnite-dimensional subsp ac e F ⊆ C M is close d in F ). Dualit y b etw ee n C M and C M . Theorem 0.16. The biline ar form h u, α i = X x ∈ M u ( x ) · α x , u ∈ C M , α ∈ C M (0.26) turns C M and C M into a dual p air h C M , C M i of ster e otyp e sp ac es: (i) every p oint char ge α ∈ C M gener ates a line ar c ontinuous functional f on C M by t he formula f ( u ) = h u, α i , u ∈ C M , and t he mapping α 7→ f is an isomorphism of lo c al ly c onvex sp ac es C M ∼ = ( C M ) ⋆ ; (ii) on the c ontr ary, every function u ∈ C M gener ates a line ar c ontinuous functional f on C M by the formula f ( α ) = h u , α i , α ∈ C M , and t he mapping u 7→ f is an isomorphism of lo c al ly c onvex sp ac es C M ∼ = ( C M ) ⋆ . Bases in C M and C M . A b asis in a top ologica l v ector space X ov er C is a family of vectors { e i ; i ∈ I } in X such tha t every vector x ∈ X can b e uniquely repr esen ted as a sum of a conv erging series in X x = X i ∈ I λ i · e i , (0.27) with coeﬃcients λ i = λ i ( x ) co n tin uously depending on x ∈ X . The s ummabilit y of series (0.27) is understo o d in the sense of Bourbaki [6]: for each neighbor hoo d of zero U in X there exists a ﬁnite set J ⊆ I suc h that for any its ﬁnite sup erset K , J ⊆ K ⊆ I x − X i ∈ I λ i · e i ∈ U (the summability of serie s (0.27) do es not mea n that this series must have ﬁnite or countable num b er o f nonzero terms). Theorem 0.17. The char acteristic funct ions { 1 x ; x ∈ G } of singletons { x } ⊆ M : 1 x ( y ) = ( 1 , x = y 0 x 6 = y , y ∈ M (0.28) form a b asis in t he top olo gic al ve ctor sp ac e C M : every funct ion u ∈ C M is a sum of a series u = X x ∈ M u ( x ) · 1 x (0.29) with c o eﬃcients u ( x ) ∈ C c ontinuously dep ending on u ∈ C M . § 1. STERE OTYPE HOPF ALGEBRAS 21 Theorem 0.18. The char acteristic funct ions of singletons { x } ⊆ M , which as elements of C M we denote by the Kr one cker symb ols { δ x ; x ∈ G } δ x ( y ) = ( 1 , x = y 0 x 6 = y , y ∈ M (0.30 ) form a b asis in t he top olo gic al ve ctor sp ac e C M : every p oint char ge α ∈ C M is a sum of a series α = X x ∈ M α x · δ x (0.31) with c o eﬃcients α x ∈ C c ontinuously dep ending on α ∈ C M . Theorem 0.19. The b ases { 1 x ; x ∈ M } (in C M ) and { δ x ; x ∈ M } (in C M ) ar e dual: h 1 x , δ y i = ( 1 , x = y 0 , x 6 = y Theorem 0.20. In the sp ac es C M and C M any t wo b ases c an b e tra nsforme d into e ach other thr ough some automorphism (i.e. a line ar home omorphism of t he sp ac e into its elf ). § 1 Stereot yp e Hopf algebras (a) T ensor pro ducts and the structure of monoidal category on Ste If X and Y are s tereot yp e spaces, then by Y : X we denote a set of all linea r contin uous mappings ϕ : X → Y endow ed with the top ology of uniform co n vergence on totally bounded sets in X . The symbol Y ⊘ X means pseudosatura tion of this space: Y ⊘ X = ( Y : X ) △ (the o pera tion of pseudos aturation △ means some sp ecial strengthening of the top ology of the initial space – see [1, § 1 ]). The space Y ⊘ X is always stereo t yp e (if X and Y are ster eot yp e). In the catego ry S te o f stereo t yp e spaces there a re tw o na tural tenso r pr oducts: – a pro jective tenso r pro duct is deﬁned by the equality X ⊛ Y = ( X ⋆ ⊘ Y ) ⋆ ; the corre sponding elementary tensor x ⊛ y ∈ X ⊛ Y ( x ∈ X , y ∈ Y ) is deﬁned by the for m ula x ⊛ y ( ϕ ) = ϕ ( y )( x ) , ϕ ∈ X ⋆ ⊘ Y ; (1.1) – an injective tensor pr oduct is deﬁned by the equality X ⊙ Y = X ⊘ Y ⋆ ; the corre sponding elementary tensor x ⊙ y ∈ X ⊙ Y ( x ∈ X , y ∈ Y ) is deﬁned by the for m ula x ⊙ y ( f ) = f ( y ) · x, f ∈ Y ⋆ . (1.2) These t wo op eratio ns are connected with each o ther by t wo isomorphis ms of functors: d : ( X ⊛ Y ) ⋆ ∼ = X ⋆ ⊙ Y ⋆ , e : ( X ⊙ Y ) ⋆ ∼ = X ⋆ ⊛ Y ⋆ 22 Theorem 1.1. The identities a ⊛ X,Y ,Z  ( x ⊛ y ) ⊛ z  = x ⊛ ( y ⊛ z ) x ∈ X , y ∈ Y , z ∈ Z (1.3) l ⊛ X ( λ ⊛ x ) = λ · x = r ⊛ X ( x ⊛ λ ) λ ∈ C , x ∈ X (1.4) c ⊛ X ( x ⊛ y ) = y ⊛ x x ∈ X , y ∈ Y (1.5) c orr e ctly deﬁne natur al isomorphisms of functors in t he c ate gory Ste a ⊛ X,Y ,Z : ( X ⊛ Y ) ⊛ Z → X ⊛ ( Y ⊛ Z ) l ⊛ X : C ⊛ X → X r ⊛ X : X ⊛ C → X c ⊛ X,Y : X ⊛ Y → Y ⊛ X These isomorphisms in t heir tu rn deﬁne a struct ur e of symmetric al monoidal c ate gory on S te [4, 23] with r esp e ct to the bifunctor ⊛ . Before formulating the next theor em let us agree to denote by h the mapping from C into C ⋆ , which every num be r λ ∈ C turns into the linear functional on C , acting as multiplication by λ : h : C → C ⋆   h ( λ )( µ ) = λ · µ ( λ , µ ∈ C ) (1.6) Clearly , h : C → C ⋆ is an isomor phism of (ﬁnite-dimensional) s tereot yp e spaces . Theorem 1.2. The formulas a ⊙ X,Y ,Z =  i X ⊙ ( i Y ⊙ i Z ) ◦ 1 X ⋆⋆ ⊙ d − 1 Y ⋆ ,Z ⋆ ◦ d − 1 X ⋆ ,Y ⋆ ⊛ Z ⋆ ◦ ( a ⊛ X ⋆ ,Y ⋆ ,Z ⋆ ) ⋆ ◦ ◦ d X ⋆ ⊛ Y ⋆ ,Z ⋆ ◦ d X ⋆ ,Y ⋆ ⊙ 1 Z ⋆⋆ ◦ ( i − 1 X ⋆ ⊙ i − 1 Y ⋆ ) ⊙ i − 1 Z ⋆  ⋆ l ⊙ X =  h − 1 ⊙ i − 1 X ◦ d C ,X ⋆ ◦ ( l ⊛ X ⋆ ) ⋆ ◦ i X  − 1 r ⊙ X =  i − 1 X ⊙ h − 1 ◦ d X ⋆ , C ◦ ( r ⊛ X ⋆ ) ⋆ ◦ i X  − 1 c ⊙ X,Y =  i − 1 X ⊙ i − 1 Y ◦ d X ⋆ ,Y ⋆ ◦ ( c ⊛ X ⋆ ,Y ⋆ ) ⋆ ◦ ( d Y ⋆ ,X ⋆ ) − 1 ◦ i Y ⊙ i X  − 1 deﬁne isomorphisms of funct ors in the c ate gory Ste a ⊙ X,Y ,Z : ( X ⊙ Y ) ⊙ Z → X ⊙ ( Y ⊙ Z ) l ⊙ X : C ⊙ X → X r ⊙ X : X ⊙ C → X c ⊙ X,Y : X ⊙ Y → Y ⊙ X such that a ⊙ X,Y ,Z  ( x ⊙ y ) ⊙ z  = x ⊙ ( y ⊙ z ) x ∈ X , y ∈ Y , z ∈ Z (1.7) l ⊙ X ( λ ⊙ x ) = λ · x = r ⊙ X ( x ⊙ λ ) λ ∈ C , x ∈ X (1.8) c ⊙ X ( x ⊙ y ) = y ⊙ x x ∈ X , y ∈ Y (1.9) These isomorphisms deﬁne a structur e of symmetric monoidal c ate gory on Ste [4] with r esp e ct t o the bifunctor ⊙ . § 1. STERE OTYPE HOPF ALGEBRAS 23 The following fact was noted in [1, T heorem 7 .9]: the identit y @ X,Y ( x ⊛ y ) = x ⊙ y , x ∈ X , y ∈ Y (1.10) deﬁnes a natural tra nsformation of the bifunctor ⊛ int o the bifunctor ⊙ @ X,Y : X ⊛ Y → X ⊙ Y , called Gr othendie ck tr ansformation . Theorem 1.3. If X and Y ar e F r´ echet sp ac es (r esp e ctively, Br auner sp ac es), then (i) the ster e otyp e tensor pr o ducts X ⊛ Y and X ⊙ Y ar e also F r´ echet sp ac es (r esp e ctively, Br auner sp ac es), (ii) if at le ast one of t he s p ac es X and Y is n ucle ar, t hen the Gr othendie ck mapping @ X,Y : X ⊛ Y → X ⊙ Y is an isomorphi sm of ster e otyp e sp ac es X ⊛ Y @ X,Y ∼ = X ⊙ Y , (iii) if b oth of the sp ac es X and Y ar e nu cle ar, then the sp ac e X ⊛ Y ∼ = X ⊙ Y is also nucle ar. Pr o of. The fa ct tha t the s paces X ⊛ Y and X ⊙ Y are F r ´ echet spa ces (r espectively , Brauner spaces), was noted in [1, Theorems 7 .22, 7.23 ]). If X or Y is nuclear, then the iso morphism X ⊛ Y ∼ = X ⊙ Y is a coro llary of the fact that for tw o F r ´ echet spaces, o f which one po ssesses the classic al appr o ximation prop erty , the tensor pr oducts ⊛ and ⊙ coincide with the usual pro jective ˆ ⊗ a nd injective ˇ ⊗ tenso r pro ducts [1, Theorems 7 .17, 7.21 ]. Finally , if b oth X and Y are nuclear, then in the ca se when b oth X and Y a re F r´ echet s paces, their tensor pro ducts X ⊛ Y ∼ = X ⊙ Y ∼ = X ˆ ⊗ Y are nuclear, since nuclearit y is inherited b y pro jectiv e tensor pro duct [25, 5 .4.2]. If X and Y are Brauner spa ces, then by what we hav e already pr o ved, the F r´ echet s pace X ⋆ ⊙ Y ⋆ is n uclear, hence, by the Brauner theorem 0.1 0 , the space X ⊛ Y ∼ = ( X ⋆ ⊙ Y ⋆ ) ⋆ is also nuclear. As a coro llary , we have Theorem 1.4. The c ate gories NF re of nucle ar F r´ echet sp ac es and NBra of nu cle ar Br auner sp ac es ar e symmetric al monoidal c ate gories with r esp e ct to bifunctors ⊛ and ⊙ (which c oincide on e ach of those c ate gories). (b) Stereot yp e Hopf algebras Algebras, coalgebras and Hopf alge bras in a symmetric mono idal category . Recall [23], that an algebr a o r a monoid in a symmetrica l mo noidal categor y ( K , ⊗ , I , a , r , l , ) ¸ is a triple ( A, µ, ι ), where A is an ob ject in K , and µ : A ⊗ A → A (multiplication) a nd ι : I → A (ident ity) a re mor phisms, sa tisfying the following axio ms of ass ociativ it y and identit y: ( A ⊗ A ) ⊗ A A ⊗ ( A ⊗ A ) A ⊗ A A A ⊗ A          µ ⊗ 1 / / a A,A,A          1 ⊗ µ / / µ o o µ I ⊗ A A A ⊗ I A ⊗ A   ? ? ? ? ? ? ? ? ? ? ι ⊗ 1 / / l A             1 ⊗ ι o o r A O O         µ . (1.11) T o any t wo monoids ( A, µ A , ι A ) and ( B , µ B , ι B ) one can assign a monoid ( A ⊗ B , µ A ⊗ B , ι A ⊗ B ) in w hic h the structure morphisms ar e deﬁned by for m ulas µ A ⊗ B := µ A ⊗ µ B ◦ θ A,B ,A,B , ι A ⊗ B := ι A ⊗ ι B ◦ l − 1 I (1.12) 24 ( A ⊗ B ) ⊗ ( A ⊗ B ) ( A ⊗ A ) ⊗ ( B ⊗ B ) A ⊗ B ) ) R R R R µ A ⊗ B / / θ A,B,A,B u u l l l l l l l l µ A ⊗ µ B I I ⊗ I A ⊗ B / / l − 1 I $ $ J J J J ι A ⊗ B z z t t t t t t ι A ⊗ ι B and θ is the iso morphism o f functors  ( A, B , C, D ) 7→ ( A ⊗ B ) ⊗ ( C ⊗ D )  θ ֌  ( A, B , C, D ) 7→ ( A ⊗ C ) ⊗ ( B ⊗ D )  (1.13) coming in as a combination of structure isomo rphisms a , l , r , c , a − 1 , l − 1 , r − 1 , c − 1 in the tensor categor y K (by the coher ence theor em [23] this morphism is unique in this cla ss). The notion of c o algebr a or c omonoid in a symmetric monoidal c ategory K is deﬁned dua lly as an arbitrar y triple ( A, κ , ε ), wher e A is a n ob ject of K , a nd κ : A → A ⊗ A (comultiplication) a nd ε : A → I (counit) ar e morphisms satisfying the dua l conditions of coasso ciativity and counit: ( A ⊗ A ) ⊗ A A ⊗ ( A ⊗ A ) A ⊗ A A A ⊗ A / / a A,A,A O O        κ ⊗ 1 A / / κ o o κ O O        1 A ⊗ κ I ⊗ A A A ⊗ I A ⊗ A o o l − 1 A / / r − 1 A           κ _ _ ? ? ? ? ? ? ? ? ? ? ε ⊗ 1 A ? ?           1 A ⊗ ε . (1.14) Like in the case of a lgebras, the tensor pro duct A ⊗ B of tw o coa lgebras ( A, κ A , ε A ) and ( B , κ B , ε B ) in K p ossesses a natural structure o f comonoid in the c ategory K w ith the s tructure mor phisms κ A ⊗ B := θ A,A,B ,B ◦ κ A ⊗ κ B , ε A ⊗ B := λ I ◦ ε A ⊗ ε B Hopf algebr a (another term – Hopf monoid ) in the s ymmetrical mono idal catego ry K is a quin tuple ( H, µ, ι, κ , ε, σ ), wher e H is an ob ject in K , and morphisms µ : H ⊗ H → H (multiplication) , ι : I → H (unit) , κ : H → H ⊗ H (comultip lication) , ε : H → I (counit) , σ : H → H (antipo de) satisfy the following co nditions: 1) the triple ( H , µ, ι ) is an algebr a in K , 2) the triple ( H , κ , ε ) is a coa lgebra in K , 3) the following diagra ms are commutativ e: H H ⊗ H H ⊗ H ( H ⊗ H ) ⊗ ( H ⊗ H ) ( H ⊗ H ) ⊗ ( H ⊗ H ) - - [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ κ ' ' O O O O O O O κ ⊗ κ 1 1 c c c c c c c c c c c c c c c c c c c c µ / / θ H,H,H,H 7 7 o o o o o o o µ ⊗ µ (1.15) I I ⊗ I H H ⊗ H           ι / / l − 1 I           ι ⊗ ι / / κ ; H ⊗ H H I ⊗ I I           ε ⊗ ε / / µ           ε / / l I ; (1.16) § 1. STERE OTYPE HOPF ALGEBRAS 25 H I I   ? ? ? ? ? ε ? ?      ι / / 1 I ; (1.17) – they mean that mo rphisms κ : H → H ⊗ H and ε : H → I ar e homomorphisms of alge bras, and morphisms µ : H ⊗ H → H and ι : I → H a re homomorphisms of coa lgebras in the catego ry K (here again θ is the tra nsformation (1.1 3)); 4) the following diagra m called antip o de axiom is commutativ e: H ⊗ H σ ⊗ 1 H / / H ⊗ H µ   = = = = = = = H κ @ @        κ   = = = = = = = ε / / I ι / / H H ⊗ H 1 H ⊗ σ / / H ⊗ H µ @ @        (1.18) If only conditions 1)-3) ar e fulﬁlled, then the quadruple ( H , µ, ι, κ , ε ) is called a bialgebr a in the categor y K . Pro jective and inje ctive stereotype algebras. In accordance with the general deﬁnition, a pr o- je ctive (resp ectively , inje ctive ) ster e otyp e algebr a is an alg ebra in the symmetric monoidal categ ory of stereotype spaces ( Ste , ⊛ ) (resp ectively , ( Ste , ⊙ )). F o r the case of pro jective algebra s this deﬁnitio n a dmits a s imple refor m ulation [1, § 10]: Prop osition 1.1. The structu r e of pr oje ctive ster e otyp e algebr a on a st er e otyp e sp ac e A is e quivalent to the structure of asso ciative (u n ital) algebr a on A , wher e the mu lt ipli c ation ( x, y ) 7→ x · y satisﬁes the fol lowing two e quivalent c onditions of c ontinuity: (i) for every c omp act set K in A and for every neighb orho o d of zer o U in A t her e exists a neighb orho o d of zer o V in A such that K · V ⊆ U & V · K ⊆ U (ii) for every c omp act set K in A and for any n et a i in A , ten ding to zer o, a i − → i →∞ 0 , the nets x · a i and a i · x ten d to zer o in A uniformly in x ∈ K . Example 1. 1. Bana c h algebr as a nd F r ´ echet alg ebras are examples of pro jective stereotype alge bras. Another example is the stereo t yp e alg ebra L ( X ) = X ⊘ X o f op erators on an arbitrar y stere ot yp e space X . Example 1.2. The standar d functiona l algebra s C ( M ), E ( M ), O ( M ), R ( M ) o f contin uous, smo oth, holomorphic functions and p olynomials ar e examples o f injective stereo t yp e alg ebras (s ee details in [1, § 10]). Stereot yp e Hopf algebras. Again, following the g eneral deﬁnition, we ca ll – a pr oje ctive st er e otyp e H opf algebr a a Hopf a lgebra in the symmetric mo noidal category of s tereot yp e spaces with the pro jective tensor pro duct ( Ste , ⊛ ); – an inje ct ive ster e otyp e Hopf algebr a a Hopf a lgebra in the sy mmetric monoida l categor y of stereotype spaces with the injective tensor pr oduct ( St e , ⊙ ); 26 – a nu cle ar Hopf-F r´ echet algebr a a Hopf algebra in the s ymmetric monoidal c ategory of n uclear F r´ echet spaces NFre ; – a nucle ar Hopf-Br auner algebr a a Hopf algebra in the symmetric monoida l category of n uclear Brauner spaces NBra . Suppo se in addition that a ster eot yp e space H is such that the Gro thendiec k transforma tion for the pair ( H ; H ), for the triple ( H ; H ; H ), and for the quadruple ( H ; H ; H ; H ) give isomorphisms of stereotype spaces: @ H,H : H ⊛ H ∼ = H ⊙ H , @ H,H,H : H ⊛ H ⊛ H ∼ = H ⊙ H ⊙ H @ H,H,H ,H : H ⊛ H ⊛ H ⊛ H ∼ = H ⊙ H ⊙ H ⊙ H (this is always the case if H is a n uclear F r´ echet spa ce or a nuclear Br auner space). Then, o b viously , every structure o f pro jective s tereot yp e Hopf algebra on H is eq uiv alent to some structure of injectiv e stereotype Hopf algebra o n H and vice versa: the structur e elements o f Hopf algebra in ( Ste , ⊛ ) a nd ( Ste , ⊙ ) (we diﬀer them by indices ⊛ and ⊙ ) either c oincide ι ⊛ = ι ⊙ , ε ⊛ = ε ⊙ , σ ⊛ = σ ⊙ or are connected by diag rams H ⊛ H H ⊙ H H   ? ? ? ? ? ? ? ? µ ⊛ / / @ H,H           µ ⊙ H ⊛ H H ⊙ H H / / @ H,H _ _ ? ? ? ? ? ? ? ? κ ⊛ ? ?         κ ⊙ W e call those (pro jective and at the sa me time injective) Hopf algebras rigid ster e otyp e Hopf algebr as . Example 1.3. Clearly , every nuclear Hopf-F r´ echet a lgebra and every nuclear Hopf-B rauner alg ebra ar e rigid stereotype Hopf a lgebras. Dualit y for ste reo t ype Hopf al g ebras. Theorem 1. 5 ( on duality for s tereot yp e Hopf alg e bras). A stru ctur e of inje ct ive (pr oje ctive, rigid) Hopf algebr a on a ster e otyp e sp ac e H automatic al ly deﬁnes a stru ctur e of pr oje ctive (inje ctive, rigid) Hopf algebr a on the dual ster e otyp e sp ac e H ⋆ – t he st ructur e elements of Hopf algebr a on H ⋆ ar e deﬁne d as the dual morphisms for the stru ctur e element s of Hopf algebr a on H : µ H ⋆ = ( κ H ) ⋆ , ι H ⋆ = ( ε H ) ⋆ , κ H ⋆ = ( µ H ) ⋆ , ε H ⋆ = ( ι H ) ⋆ , σ H ⋆ = ( σ H ) ⋆ . Example 1.4. Every Hopf algebra in usual sense H (i.e. a Hopf a lgebra in the ca tegory of vector spaces with the usua l alge braic tensor pro duct ⊗ ) b ecomes a r igid Hopf alg ebra, b eing endow ed w ith the strongest lo cally conv ex top ology . The dua l space H ⋆ with resp ect to this top ology is a space of minimal t yp e (in the sense of Theorem 0.12) and a rigid Ho pf a lgebra, as well as H . This, by the wa y , illus trates one of the adv a n tages o f stereotype theory : here we do not need to narrow the space of linear functionals to make a Hopf a lgebra fro m them, like it is usually done (see e.g . [9, 1.5 ] or [8, 4 .1.D]) – the s pace H ⋆ , being a space of all linear functionals (automatically cont inuous, by the choice of the top ology in H ), is a “true” Hopf alg ebra, but to s ee this we hav e to ta k e one o f the stereotype tensor pr oducts ⊛ or ⊙ instead of the alge braic tensor pro duct ⊗ . § 1. STERE OTYPE HOPF ALGEBRAS 27 Dual pairs. Let H b e an injective, and M a pro jective stereo t yp e Hopf a lgebras. Let us say that H and M fo rm a dual p air of Hopf algebr as , if there is a non-degener ate co n tin uous (in the sense of [1, Section 5.6 ]) bilinear form h· , ·i : H × M → C , which turns algebraic op erations in H into dual op erations in M : h µ H ( U ) , α i = h U, κ M ( α ) i , h 1 H , α i = ε A ( α ) , h κ H ( u ) , A i = h u, µ M ( A ) i , ε H ( u ) = h u, 1 M i , h σ H ( u ) , α i = h u, σ M ( α ) i ( u ∈ H , U ∈ H ⊙ H , α ∈ M , A ∈ M ⊛ M ). If h H , M i is a dual pair of stere ot yp e Hopf alg ebras, where H is an injective, M a pro jective Hopf algebra , then on default we sha ll denote by dot · the m ultiplication in H , by snowﬂak e ∗ the multiplication in M : µ H ( u ⊙ v ) = u · v , µ M ( α ⊛ β ) = α ∗ β Example 1 .5. Certa inly , the pair h H , H ⋆ i , where H is an injective stereptype Hopf algebra, and h· , ·i the canonical bilinear for m, h a, α i := α ( a ) , a ∈ H , α ∈ H ⋆ is a n example of dual pair of Ho pf algebra s. Below we use the notation h· , ·i for this form without supplement ar y explana tions. (c) Key example: Hopf algebras C G and C G Recall that in § 0(h) we deﬁned the spaces C M and C M of functions and of po in t charges on a set M . If M is a g roup, then the spaces C M and C M naturally turn into a dual pair o f Hopf alge bras. Algebra C G of functions o n G . Let G b e an arbitra ry group (no t necess arily ﬁnite) a nd let C G be the space of all (complex- v alued) functions on G , deﬁned in § 0(h), u ∈ C G ⇐ ⇒ u : G → C with the to polog y o f po in twise conv erge nce (genera ted by semino rms (0.23)). W e endow C G with the supplement ar y structure o f alg ebra with the p oint wise algebra ic o pera tions: ( u · v )( x ) = u ( x ) · v ( x ) , 1 C G ( x ) = 1 ( x ∈ G ) . (1.19) Recall that by for m ula (0 .28) ab o ve we deﬁned the characteristic functions 1 x of singletons in G . The m ultiplication in C G can b e written in the decomp osition by the ba sis { 1 x ; x ∈ G } by formula u · v = X x ∈ G u ( x ) · 1 x ! · X x ∈ G v ( x ) · 1 x ! = X x ∈ G u ( x ) · v ( x ) · 1 x (1.20) and on the elements of this basis lo oks as follows: 1 x · 1 y = ( 1 x , x = y 0 x 6 = y (1.21) The algebra C G of p oin t c harges on G . Again let G b e a n ar bitrary gr oup and C G the space of po in t charges on G , deﬁned in § 0(h), α ∈ C G ⇐ ⇒ α = { α x ; x ∈ G } , α x ∈ C , card { x ∈ G : α x 6 = 0 } < ∞ 28 W e endow C G with the top ology genera ted by seminor ms (0.25) (by theorem 0 .14, this is equiv alent to the stro ngest lo cally conv ex top ology on C G ). Besides this C G is endow ed with the str ucture of algebr a under the multiplication ( α ∗ β ) y = X x ∈ G α x · β x − 1 · y By formula (0.30) we deﬁned the characteristic function o f singletons in G , w hic h, b eing c onsidered a s elements in C G were denoted by the symbols of delta- functionals: δ x . The unit in C G is the characteristic function δ e suppo rted in the unit e of the gr oup G : δ e x = ( 1 , x = e 0 x 6 = e The m ultiplication in the algebra C G is written in the decomp osition b y elemen ts of the basis { δ x ; x ∈ G } by formula α ∗ β = X x ∈ G α x · δ x ! ∗   X y ∈ G β y · δ y   = X x,y ∈ G α x · β y · δ x · y = X z ∈ G X x ∈ G α x · β x − 1 · z ! · δ z (1.22) and on the elements of this basis lo oks as follows: δ x ∗ δ y = δ x · y . (1.23) C G and C G as stereo type Hopf algebras. F or tw o ar bitrary sets S and T and for tw o functions u : S → C and v : T → C let the symbol u ⊡ v deno te the function on the Ca rtesian pro duct S × T deﬁned by the identit y ( u ⊡ v )( s, t ) := u ( s ) · v ( t ) , s ∈ S , t ∈ T (1.24) The deﬁnition of the Hopf algebra in the inﬁnite-dimensional alge bras C G and C G is ba sed on the followin g observ ation: Theorem 1.6. The formula ρ S,T ( u ⊡ v ) = u ⊙ v (1.25) deﬁnes an isomorphism of t op olo gic al ve ctor sp ac es ρ S,T : C S × T ∼ = C S ⊙ C T This isomorphism is an isomorphism of functors,  ( S ; T ) 7→ C S × T  ρ S,T ֌  ( S ; T ) 7→ C S ⊙ C T  , sinc e for any mappings π : S → S ′ and σ : T → T ′ the fol lowing diagr am is c ommutative: C S × T C S ⊙ C T C S ′ × T ′ C S ′ ⊙ C T ′ / / ρ S,T O O id C ⊘ ( π × σ ) / / ρ S ′ ,T ′ O O ( id C ⊘ π ) ⊙ ( id C ⊘ σ ) – her e t he mappings id C ⊘ ? ar e deﬁne d by formula id C ⊘ π : C S ′ → C S , id C ⊘ π ( v ) = v ◦ π § 1. STERE OTYPE HOPF ALGEBRAS 29 This theorem allows to deﬁne the structure e lemen ts of Hopf alg ebra o n C G with res pect to ⊙ : the mapping µ : C G ⊙ C G → C G is initially deﬁned on the set of functions on the Cartesia n pr oduct G × G , e µ : C G × G → C G    e µ ( v )( t ) = v ( t, t ) and then is extended to tensor square by the iso morphism ρ G,G : µ = e µ ◦ ρ G,G . Similarly the comultiplication κ : C G → C G ⊙ C G is initially deﬁned with the v alues in the space of functions on the Ca rtesian squa re e κ : C G → C G × G    e κ ( u )( s, t ) = u ( s · t ) and after that is extended to tensor squa re by the isomor phism ρ G,G : κ = ρ G,G ◦ e κ . The other structure elements of Hopf algebra on C G are o b vious: unit: ι : C → C G , ι ( λ )( t ) = λ counit: ε : C G → C , ε ( u ) = u (1 G ) antipo de: σ : C G → C G , σ ( u )( t ) = u ( t − 1 ) The following picture illustra tes these deﬁnitions: C G × G C C G C G C C G ⊙ C G $ $ J J J J J J e µ   ρ G,G / / ι : : t t t t t t e κ $ $ J J J J J J κ / / ε : : t t t t t t µ (1.26) On the space C G of p oint charges on G the s tructure of Hopf algebra with resp ect to tenso r pro duct ⊛ is deﬁned dually by Theorem 1 .5, as on the dual space to C G in the sens e o f bilinear for m (0.2 6). The following theorem shows that these deﬁnitions indeed deﬁne a structure of Hopf a lgebra on C G and C G : Theorem 1.7. F or any gr oup G – the sp ac e C G of functions on G is an inje ctive (and m or e over, a rigid) st ere otyp e Hopf algebr a; if in addition G is c ountable, then C G is a nucle ar Hopf-F r´ echet algebr a; – the sp ac e C G of p oint char ges on G is a pr oje ctive (and mor e over, a rigid) ster e otyp e Hopf algebr a; if in addition G is c ountable, t hen C G is a nucle ar Hopf-Br auner algebr a. The Hopf algebr as C G and C G form a dual p air with r esp e ct to biline ar form (0.26) , and the algebr aic op er ations on t hem act on b ases { 1 x } and { δ x } by formulas: C G : 1 x · 1 y = ( 1 x , x = y 0 , x 6 = y 1 C G = X x ∈ G 1 x σ (1 x ) = 1 x − 1 (1.27) κ (1 x ) = X y ∈ G 1 y ⊙ 1 x · y − 1 ε (1 x ) = ( 1 , x = e 0 , x 6 = e (1.28) C G : δ x ∗ δ y = δ x · y 1 C G = δ e σ ( δ x ) = δ x − 1 (1.29) κ ( δ x ) = δ x ⊛ δ x ε ( δ x ) = 1 (1.30) 30 Pr o of. F o rm ulas (1.27)-(1.30) are veriﬁed by direct calculation. The r igidit y follows from [1, (7.3 7)]: C I ⊛ C J ∼ = C I × J ∼ = C I ⊙ C J , C I ⊛ C J ∼ = C I × J ∼ = C I ⊙ C J . The structure of Hopf algebra on C G is generated b y the structure of Hopf algebra on C G due to Theorem 1.5. T h us, we need only to prove that C G is a Hopf algebra with resp ect to ⊙ . 1. Let us chec k diag ram (1.15). Replace ev erywher e H by C G and ⊗ by ⊙ , and after that let us ov erbuild this diagra m to the following prism: C G C G × G C G × G C ( G × G ) × ( G × G ) C ( G × G ) × ( G × G ) C G × G ⊙ C G × G C G × G ⊙ C G × G C G C G ⊙ C G C G ⊙ C G  C G ⊙ C G  ⊙  C G ⊙ C G   C G ⊙ C G  ⊙  C G ⊙ C G  + + W W W W W W W W W W W W W W W W W W W W W e κ   ρ G,G ' ' O O O O O O O e e κ 3 3 g g g g g g g g g g g g g g g g g g g g g e µ   ρ G,G / / e θ   ρ G × G,G × G   id C G 7 7 o o o o o o o e e µ   ρ G × G,G × G   ρ G,G ⊙ ρ G,G   ρ G,G ⊙ ρ G,G # # G G G G G G G G G G G e µ ⊙ e µ W W W W W W W W W W 3 3 g g g g g g g g g g µ + + W W W W W W W W κ ; ; w w w w w w w w w w w e κ ⊙ e κ ' ' O O O O O O κ ⊙ κ g g g g g g g g / / θ 7 7 o o o o o o µ ⊙ µ Here θ = θ C G , C G , C G , C G is the is omorphism of functors from (1.1 3), and the other mor phisms ar e deﬁned as follows: e θw ( a, b, c, d ) = w ( a, c, b, d ) , e e κ v ( a, b, c, d ) = v ( a · b , c · d ) , e e µv ( a, b ) = v ( a, a, b, b ) T o prov e that the base of the pr ism is commutativ e, it is suﬃcient to verify that a ll the other faces ar e commutativ e. The r emote later al faces C G × G C G C G ⊙ C G C G            ρ G,G / / e µ            id C G / / µ C G C G × G C G C G ⊙ C G            id C G / / e κ            ρ G,G / / κ – are just distorted triangles from diagra m (1 .26). In the left nearby face C G × G C ( G × G ) × ( G × G ) C G × G ⊙ C G × G C G ⊙ C G  C G ⊙ C G  ⊙  C G ⊙ C G                       ρ G,G / / e e κ           ρ G × G,G × G           ρ G,G ⊙ ρ G,G 7 7 o o o o o o o o o o o o o o o o o e κ ⊙ e κ / / κ ⊙ κ § 1. STERE OTYPE HOPF ALGEBRAS 31 – the low er tr iangle is just left triang le in (1 .26) mult iplied by itse lf via the op eration ⊙ , a nd the com- m utativity of the inner quadrang le is veriﬁed by the substituting the function u ⊡ v ∈ C G × G , u, v ∈ C G , as an ar gumen t: if we move down, a nd then r igh t and up, we turn this function into e κ ( u ) ⊙ e κ ( v ), u ⊡ v ∈ C G × G 7→ u ⊙ v ∈ C G ⊙ C G 7→ ( e κ ⊙ e κ )( u ⊙ v ) = e κ ( u ) ⊙ e κ ( v ) ∈ C G × G ⊙ C G × G – and if we mov e right and then down, we obtain the same result: u ⊡ v ∈ C G × G 7→ e e κ ( u ⊡ v ) = e κ ( u ) ⊡ e κ ( v ) ∈ C ( G × G ) × ( G × G ) : e e κ ( u ⊡ v )( a, b, c, d ) = ( u ⊡ v )( a · b, c · d ) = u ( a · b ) · v ( c · d ) = e κ ( u )( a, b ) · e κ ( v )( c, d ) = =  e κ ( u ) ⊡ e κ ( v )  ( a, b, c, d ) 7→ ρ G × G,G × G ( e κ ( u ) ⊡ e κ ( v )) = (1.25) = e κ ( u )) ⊙ e κ ( v ) ∈ C G × G ⊙ C G × G . The commutativit y of the central near b y face: C ( G × G ) × ( G × G ) C ( G × G ) × ( G × G ) C G × G ⊙ C G × G C G × G ⊙ C G × G  C G ⊙ C G  ⊙  C G ⊙ C G   C G ⊙ C G  ⊙  C G ⊙ C G    ρ G × G,G × G / / e θ   ρ G × G,G × G   ρ G,G ⊙ ρ G,G   ρ G,G ⊙ ρ G,G / / θ – is veriﬁed by substituting a function ( u ⊡ v ) ⊡ ( p ⊡ q ) ∈ C G × G as an a rgument, its motion due to (1.25) will b e as follows: ( u ⊡ v ) ⊡ ( p ⊡ q ) _ ρ G × G,G × G    e θ / / ( u ⊡ p ) ⊡ ( v ⊡ q ) _ ρ G × G,G × G   ( u ⊡ v ) ⊙ ( p ⊡ q ) _ ρ G,G ⊙ ρ G,G   ( u ⊡ p ) ⊙ ( v ⊡ q ) _ ρ G,G ⊙ ρ G,G   ( u ⊙ v ) ⊙ ( p ⊙ q )  θ / / ( u ⊙ p ) ⊙ ( v ⊙ q ) In the right near b y face C ( G × G ) × ( G × G ) C G × G C G × G ⊙ C G × G  C G ⊙ C G  ⊙  C G ⊙ C G  C G ⊙ C G           ρ G × G,G × G / / e e µ                      ρ G,G           ρ G,G ⊙ ρ G,G ' ' O O O O O O O O O O O O O O O O O e µ ⊙ e µ / / µ ⊙ µ 32 – the low er tr iangle is just the right tria ngle in (1.2 6) m ultiplied b y itself via the op eration ⊙ , and the commutativit y of the inner quadrangle is veriﬁed by taking as an a rgument the function u ⊡ v ∈ C ( G × G ) × ( G × G ) , u , v ∈ C G × G : if we mov e down, and then right and down, we o btain u ⊡ v ∈ C ( G × G ) × ( G × G ) 7→ ρ G × G,G × G ( u ⊡ v ) = u ⊙ v ∈ C G × G ⊙ C G × G 7→ ( e µ ⊙ e µ )( u ⊙ v ) = e µ ( u ) ⊙ e µ ( v ) ∈ C G ⊙ C G – and if we mov e right and then down, we obtain the same result: u ⊡ v ∈ C ( G × G ) × ( G × G ) 7→ e e µ ( u ⊡ v ) = e µ ( u ) ⊡ e µ ( v ) ∈ C G × G : e e µ ( u ⊡ v )( a, b ) = ( u ⊡ v )( a, a, b , b ) = u ( a, a ) · v ( b · b ) = e µ ( u )( a ) · e µ ( v )( b ) =  e µ ( u ) ⊡ e µ ( v )  ( a, b ) 7→ ρ G,G ( e µ ( u ) ⊡ e µ ( v )) = (1.25) = e µ ( u )) ⊙ e µ ( v ) ∈ C G ⊙ C G . It remains to chec k the commutativit y of the upper base of the prism. C G C G × G C G × G C ( G × G ) × ( G × G ) C ( G × G ) × ( G × G ) + + W W W W W W W W W W W W W W e κ ' ' O O O e e κ 3 3 g g g g g g g g g g g g g g e µ / / e θ 7 7 o o o e e µ A function v ∈ C G × G , being moved thro ugh the upper tw o edges of the pentagon, undergo es the following transmutations: v ∈ C G × G 7→ e µv ∈ C G , e µv ( a ) = v ( a, a ) 7→ e κ ( e µv ) ∈ C G × G , e κ ( e µv ) ( a, b ) = e µv ( a · b ) = v ( a · b , a · b ) – and the res ult is the sa me as if we move it through the three low er edges: v ∈ C G × G 7→ e e κ v ∈ C ( G × G ) × ( G × G ) , e e κ v ( a, b, c, d ) = v ( a · b, c · d ) 7→ e θ  e e κ v  ∈ C ( G × G ) × ( G × G ) , e θ  e e κ v  ( a, b, c, d ) = e e κ v ( a, c, b, d ) = v ( a · c, b · d ) 7→ e e µ  e θ  e e κ v  ∈ C G × G , e e µ  e θ  e e κ v  ( a, b ) = e θ  e e κ v  ( a, a, b, b ) = e e κ v ( a, b, a, b ) = v ( a · b, a · b ) 2. After tha t we verify diagrams (1.16), which fo r the a lgebra C G come to the following form: C ι   l − 1 C / / C ⊙ C ι ⊙ ι   C G κ / / C G ⊙ C G C G ⊙ C G ε ⊙ ε   µ / / C G ε   C ⊙ C l C / / C Let us input an arbitrar y num ber ζ ∈ C as a argument into the ﬁrst diagra m, and an elementary tensor u ⊙ v ∈ C G ⊙ C G int o the seco nd diagr am, and apply identities (1.8): ζ = ζ · 1 C _ ι    l − 1 C / / ζ · 1 C ⊙ 1 C _ ι ⊙ ι   ζ · 1 C G  κ / / ζ · 1 C G ⊙ 1 C G u ⊙ v _ ε ⊙ ε    µ / / u · v _ ε   u (1 G ) ⊙ v (1 G )  l C / / u (1 G ) · v (1 G ) § 1. STERE OTYPE HOPF ALGEBRAS 33 3. It r emains to verify diagra m (1.1 8), which for C G comes to the for m: C G ⊙ C G σ ⊙ 1 C G / / C G ⊙ C G µ   : : : : : : : : : : C G κ B B           κ   : : : : : : : : : : ε / / C ι / / C G C G ⊙ C G 1 C G ⊙ σ / / C G ⊙ C G µ B B           (1.31) W e ov erbuild it to the diagr am C G × G ρ G,G   σ 1 / / C G × G ρ G,G   e µ   C G ⊙ C G σ ⊙ 1 C G / / C G ⊙ C G µ   : : : : : : : : : : C G e κ @ @ e κ   κ B B           κ   : : : : : : : : : : ε / / C ι / / C G C G ⊙ C G 1 C G ⊙ σ / / C G ⊙ C G µ B B           C G × G ρ G,G O O σ 2 / / C G × G ρ G,G O O e µ O O where the mappings σ 1 and σ 2 are de ﬁned by the iden tities σ 1 ( v )( s, t ) := v ( s − 1 , t ) , σ 2 ( v )( s, t ) := v ( s, t − 1 ) Obviously , all the triangles a nd q uadrangles siding with the b orders o f this picture, are comm utative here. Hence, to prov e the co mm utativity of the tw o inner p en tago ns (i.e. the co mm utativity of (1.31)) it is suﬃcient to chec k the c omm utativity o f the diagram ar ising a fter throwing o ut the vertexes C G ⊙ C G : C G × G σ 1 / / C G × G e µ   C G e κ > > e κ ε / / C ι / / C G C G × G σ 2 / / C G × G e µ H H This is done by the direct calc ulation: for any function u ∈ C G its ﬁnal image after mo tion through the diagram is the function u (1 G ) · 1 C G ∈ C G . Indee d, 34 – if we mov e through the upp er a rrows, we obtain the following: u ∈ C G 7→ e κ ( u ) ∈ C G × G , e κ ( u )( s, t ) = u ( s · t ) 7→ σ 1 ( e κ ( u )) ∈ C G × G , σ 1 ( e κ ( u ))( s, t ) = e κ ( u )( s − 1 , t ) = u ( s − 1 · t ) 7→ e µ  σ 1 ( e κ ( u ))  ∈ C G , e µ  σ 1 ( e κ ( u ))  ( s ) = σ 1 ( e κ ( u ))( s, s ) = e κ ( u )( s − 1 , s ) = u ( s − 1 · s ) = u (1 G ) – if we mov e aﬂat through the center o f the p en tago n, we obtain the same: u ∈ C G 7→ ε ( u ) ∈ C , ε ( u ) = u (1 G ) · 1 C 7→ ι ( ε ( u )) ∈ C G , ι ( ε ( u )) = u (1 G ) · 1 C G – and if we move throug h the low er arrows, we come to the sa me res ult: u ∈ C G 7→ e κ ( u ) ∈ C G × G , e κ ( u )( s, t ) = u ( s · t ) 7→ σ 2 ( e κ ( u )) ∈ C G × G , σ 2 ( e κ ( u ))( s, t ) = e κ ( u )( s, t − 1 ) = u ( s · t − 1 ) 7→ e µ  σ 2 ( e κ ( u ))  ∈ C G , e µ  σ 2 ( e κ ( u ))  ( s ) = σ 1 ( e κ ( u ))( s, s ) = e κ ( u )( s, s − 1 ) = u ( s · s − 1 ) = u (1 G ) (d) Sw eedler’s notations and the stereot yp e appro ximation prop erty A useful instrument for proving results in the theory of Hopf alg ebra a re Sweedler’s no tations [19]. This techn ique can b e applied in the stereo t yp e theor y as well, at least in situatio ns, where a given stereo t yp e Hopf algebr a H , b eing co nsidered a s a s tereot yp e space p osses ses the ster eot yp e a pproximation proper t y (see [1]). The following result explains this: Theorem 1 .8. If H is an inje ctive ( r esp e ct ively , a pr oje ctive) ster e otyp e c o algebr a with t he ster e otyp e appr oximation pr op erty, t hen for any x ∈ H (i) the c omultiplic ation κ ( x ) c an b e appr oximate d in the t op olo gy of H by the ﬁnite sums of the form n X i =1 x ′ i ⊙ x ′′ i n X i =1 x ′ i ⊛ x ′′ i ! (ii) the identity h κ ( x ) , α ⊛ β i = 0  h κ ( x ) , α ⊙ β i = 0  , α, β ∈ H ⋆ is e quivalent to t he identity κ ( x ) = 0 Pr o of. This follows from the deﬁnition of the stere ot yp e appr o ximation [1, § 9]. If now H , s a y , is an injective s tereotype coalg ebra with the ster eot yp e a pproximation prop erty , then for each e lemen t x ∈ H the sy m bo l P ( x ) x ′ ⊙ x ′′ denotes the class of nets of the form P n ν i =1 x ′ ν,i ⊙ x ′′ ν,i tending to κ ( x ): κ ( x ) ← − ∞← ν n ν X i =1 x ′ ν,i ⊙ x ′′ ν,i (1.32) § 1. STERE OTYPE HOPF ALGEBRAS 35 and the reco rd κ ( x ) = X ( x ) x ′ ⊙ x ′′ (1.33) should b e unders too d as follows: the right s ide denotes one o f thos e nets (but without indices), and the arrow is replace d by the equality . The formulas for comultiplication a nd a n tipo de, and the o thers, like κ ( σ ( x )) = X ( x ) σ ( x ′′ ) ⊙ σ ( x ′ ) are interpreted a s follows: for every net P n ν i =1 x ′ ν,i ⊙ x ′′ ν,i the condition (1.32) automatically implies the condition κ ( σ ( x )) ← − ∞← ν n ν X i =1 σ ( x ′′ ν,i ) ⊙ σ ( x ′ ν,i ) The pro of can also b e conducted with the help of the record (1.33): h κ ( σ ( x )) , α ⊛ β i = h σ ( x ) , α ∗ β i = h x, σ ⋆ ( α ∗ β ) i = h x, σ ⋆ ( β ) ∗ σ ⋆ ( α ) i = h κ ( x ) , σ ⋆ ( β ) ⊛ σ ⋆ ( α ) i = = * X ( x ) x ′ ⊙ x ′′ , σ ⋆ ( β ) ⊛ σ ⋆ ( α ) + = X ( x ) h x ′ ⊙ x ′′ , σ ⋆ ( β ) ⊛ σ ⋆ ( α ) i = X ( x ) h x ′ , σ ⋆ ( β ) i · h x ′′ , σ ⋆ ( α ) i = = X ( x ) h σ ( x ′ ) , β i · h σ ( x ′′ ) , α i = X ( x ) h σ ( x ′′ ) ⊙ σ ( x ′ ) , α ⊛ β i = * X ( x ) σ ( x ′′ ) ⊙ σ ( x ′ ) , α ⊛ β + As an exa mple of the application of these “gene ralized” Sweedler’s nota tions let us consider the following situation. In the theor y of q uan tum groups the veriﬁcation of the diag ram for a n tip ode (1.18), i.e. the iden tity µ  ( σ ⊗ 1 )( κ ( x ))  = ε ( x ) · 1 H = µ  (1 ⊗ σ )( κ ( x ))  , x ∈ H (1.34) often leads to some bulky co mputations. In those c ases one remark, made b y A. V an Daele in [39] is useful. Being applied to stere ot yp e algebra s it lo oks as follows: Lemma 1.1. Supp ose H is a ster e otyp e bialgebr a (no matter, pr oje ctive, or inje ctive) with t he ster e otyp e appr oximation pr op erty, σ is its (c ontinuous) antihomomorphism and the e qualities (1.34) ar e true for two elements x ∈ H and y ∈ H . Then they ar e true for their mu lt ipli c ation x · y . Pr o of. Both those equalities for x · y are pr o ved b y direc t computations, for instance, the left one is obtained as follows (here the tensor pro duct ⊗ means ⊛ or ⊙ ): µ  ( σ ⊗ 1 )( κ ( x · y ))  = µ  ( σ ⊗ 1 )( κ ( x ) · κ ( y ))  = µ   ( σ ⊗ 1 )   X ( x ) x ′ ⊗ x ′′ · X ( y ) y ′ ⊗ y ′′     = = X ( x ) , ( y ) µ  ( σ ⊗ 1) ( x ′ · y ′ ⊗ x ′′ · y ′′ )  = X ( x ) , ( y ) µ  σ ( x ′ · y ′ ) ⊗ x ′′ · y ′′  = X ( x ) , ( y ) µ  σ ( y ′ ) · σ ( x ′ ) ⊗ x ′′ · y ′′  = = X ( x ) , ( y ) σ ( y ′ ) · σ ( x ′ ) · x ′′ · y ′′ = X ( y ) σ ( y ′ ) ·   X ( x ) σ ( x ′ ) · x ′′   · y ′′ = X ( y ) σ ( y ′ ) · ε ( x ) · 1 H · y ′′ = = ε ( x ) · 1 H · X ( y ) σ ( y ′ ) · y ′′ = ε ( x ) · 1 H · ε ( y ) · 1 H · = ε ( x · y ) · 1 H · 36 (e) Grouplik e elemen ts Prop osition 1 .2. F or an element a ∈ H in a ster e otyp e Hopf algebr a H with the ster e otyp e appr oximation pr op erty the fol lowing c onditions ar e e quivalent: (i) κ ( a ) = a ⊙ a ( κ ( a ) = a ⊛ a ) ; (ii) the functional h a, ·i : H ⋆ → C is multiplic ative: h a, α ∗ β i = h a, α i · h a, β i (1.35) (iii) the op er ator M ⋆ a : H ⋆ → H ⋆ dual to the op er ator of multiplic ation by element a , M a ( x ) := a · x is a homomorphi sm of ster e otyp e algebr a H ⋆ . Pr o of. Obviously , (i) and (ii) are equiv alent. Let us show that (i) ⇐ ⇒ (iii). If a satisﬁes (i), then h x, M ⋆ a ( α ∗ β ) i = h M a ( x ) , α ∗ β i = h a · x, α ∗ β i = h κ ( a · x ) , α ⊛ β i = h κ ( a ) · κ ( x ) , α ⊛ β i = = * a ⊙ a · X ( x ) x ′ ⊙ x ′′ , α ⊛ β + = * X ( x ) ( a · x ′ ) ⊙ ( a · x ′′ ) , α ⊛ β + = X ( x ) h a · x ′ , α i · h a · x ′′ , β i = = X ( x ) h x ′ , M ⋆ a ( α ) i · h x ′′ , M ⋆ a ( β ) i = * X ( x ) x ′ ⊙ x ′′ , M ⋆ a ( α ) ⊛ M ⋆ a ( β ) + = h κ ( x ) , M ⋆ a ( α ) ⊛ M ⋆ a ( β ) i = = h x, M ⋆ a ( α ) ∗ M ⋆ a ( β ) i This is true for every x ∈ H , ther efore M ⋆ a ( α ∗ β ) = M ⋆ a ( α ) ∗ M ⋆ a ( β ) (1.36) On the contrary , if (1.36) is true, then h κ ( a ) , α ⊛ β i = h a, α ∗ β i = h 1 , M ⋆ a ( α ∗ β ) i = h 1 , M ⋆ a ( α ) ∗ M ⋆ a ( β ) i = h κ (1) , M ⋆ a ( α ) ⊛ M ⋆ a ( β ) i = = h 1 ⊙ 1 , M ⋆ a ( α ) ⊛ M ⋆ a ( β ) i = h 1 , M ⋆ a ( α ) i · h 1 , M ⋆ a ( β ) i = h a, α i · h a, β i = h a ⊙ a, α ⊛ β i and since this is tr ue for each α and β , we obtain (i). An element a o f an injective (resp., pro jective) stereotype Hopf a lgebra H is called gr ouplike element , if a 6 = 0 a nd a satisﬁes the conditions (i)-(ii)-(iii) of Pr opos ition 1.2. The set of gr ouplik e elements in H is denoted by G ( H ). As in the pure algebr aic situation (see [3 7]), G ( H ) is a g roup with resp ect to the m ultiplication in H , since it p ossesses the following pro perties: 1 ◦ . ∀ a ∈ G ( H ) ε ( a ) = 1 . 2 ◦ . ∀ a ∈ G ( H ) a − 1 = σ ( a ) ∈ G ( H ) . 3 ◦ . ∀ a, b ∈ G ( H ) a · b ∈ G ( H ) . Pr o of. 1 ◦ is prov ed by applying the comultiplication ax iom: 1 ⊙ a = l − 1 H ( a ) = (1.14) = ( ε ⊙ id H )( κ ( a )) = ( ε ⊙ id H )( a ⊙ a ) = ε ( a ) ⊙ a = ⇒ ε ( a ) = 1 In 2 ◦ we need to apply the fact t hat σ ⋆ is an antihomomorphism: on the one h and, κ ( σ ( a )) = σ ( a ) ⊙ σ ( a ), since § 2. STEIN MANIFOLDS: RECT ANGLES IN O ( M ) AND RHOMBUSES IN O ⋆ ( M ) 37 h κ ( σ ( a )) , α ⊛ β i = h σ ( a ) , α ∗ β i = h a, σ ⋆ ( α ∗ β ) i = h a, σ ⋆ ( β ) ∗ σ ⋆ ( α ) i = h κ ( a ) , σ ⋆ ( β ) ⊛ σ ⋆ ( α ) i = = h a ⊙ a, σ ⋆ ( β ) ⊛ σ ⋆ ( α ) i = h a, σ ⋆ ( β ) i · h a, σ ⋆ ( α ) i = h σ ( a ) , β i · h σ ( a ) , α i = h σ ( a ) ⊙ σ ( a ) , α ⊛ β i And, on the other hand, ε ( σ ( a )) = h σ ( a ) , 1 H i = h a, σ ⋆ (1 H ) i = h a, 1 H i = ε ( a ) = 1. T ogether these conditions mean that σ ( a ) ∈ G ( H ). Apart from that, σ ( a ) · a = µ ( σ ( a ) ⊙ a ) = µ  ( σ ⊙ id H )( a ⊙ a )  = µ  ( σ ⊙ id H )( κ ( a ))  = (1.18) = ε ( a ) · 1 H = (prop erty 1 ◦ ) = 1 H and, similarly , a · σ ( a ) = 1 H . Hence , σ ( a ) = a − 1 . Finally , 3 ◦ : if a , b ∈ G ( H ), then, ﬁrst, κ ( a · b ) = κ ( a ) · κ ( b ) = a ⊙ a · b ⊙ b = ( a · b ) ⊙ ( a · b ), and, second, ε ( a · b ) = ε ( a ) · ε ( b ) = 1. Ther efore, a · b ∈ G ( H ). Recall that an element a of a n algebra A is called c entr al , if it commutes with all other elements of A : ∀ x ∈ A a · x = x · a Prop osition 1.3. If a is a gr ouplike and in addition a c entr al element in a Hopf algebr a H , t hen 1) the fol lowing identities hold: M a ◦ σ ◦ M a = σ, σ ◦ M a = M a − 1 ◦ σ (1.37) M ⋆ a ◦ σ ⋆ ◦ M ⋆ a = σ ⋆ , σ ⋆ ◦ M ⋆ a = M ⋆ a − 1 ◦ σ ⋆ (1.38) 2) if H has the ster e otyp e appr oximation pr op ert y, t hen the fol lowing identities hold: κ  M ⋆ a ( α )  = X ( α ) M ⋆ a ( α ′ ) ⊛ α ′′ = X ( α ) α ′ ⊛ M ⋆ a ( α ′′ ) , α ∈ H ⋆ (1.39) κ  ( M ⋆ a ) i + j ( α )  = X ( α ) ( M ⋆ a ) i ( α ′ ) ⊛ ( M ⋆ a ) j ( α ′′ ) , α ∈ H ⋆ , i , j ∈ N (1.40) Pr o of. 1. F or each x ∈ H we have ( M a ◦ σ ◦ M a )( x ) = a · σ ( a · x ) = a · σ ( x ) · σ ( a ) = a · σ ( x ) · a − 1 = σ ( x ) · a · a − 1 = σ ( x ) 2. F o r a n y u, v ∈ H , α ∈ H ⋆ h u ⊙ v , κ ( M ⋆ a ( α )) i = h u · v , M ⋆ a ( α ) i = h a · u · v , α i = h ( a · u ) ⊙ v , κ ( α ) i = * ( a · u ) ⊙ v , X ( α ) α ′ ⊛ α ′′ + = = X ( α ) h ( a · u ) ⊙ v, α ′ ⊛ α ′′ i = X ( α ) h ( a · u ) , α ′ i · h v, α ′′ i = X ( α ) h u, M ⋆ a ( α ′ ) i · h v , α ′′ i = X ( α ) h u ⊙ v , M ⋆ a ( α ′ ) ⊛ α ′′ i = = * u ⊙ v, X ( α ) M ⋆ a ( α ′ ) ⊛ α ′′ + By Theo rem 1.8 this means that the ﬁrst equality in (1.39) holds. The rest equalities are prov ed b y analogy . § 2 Stein manifolds: rectangles in O ( M ) and rhom buses in O ⋆ ( M ) Here we discuss some sp ecial prop erties of the space of holomorphic functions on a co mplex ma nifold. F o r illustra tion pur poses it is conv enient for us to use the condition of holomor phic separa bilit y , so we formulate our res ults only for Stein manifolds. W e use terminology from [34, 38, 1 4]. 38 (a) Stein manifolds Let M b e a complex manifold. Symbol O ( M ) deno tes the algebra of a ll holomor phic functions on M (with the usual p oint wise alg ebraic op erations and top ology of uniform co n vergence o n compact sets in M ). It is well-known ( se e [38]), that a s a top ologica l vector spac e, O ( M ) is a Montel space. A manifold M is c alled a Stein manifold [34], if the fo llo wing thr ee conditions are fulﬁlled: 1) holomorph ic sep ar ability: for any tw o p oints x, y ∈ M , x 6 = y , there exists a function u ∈ O ( M ) such that u ( x ) 6 = u ( y ) 2) holomorph ic un ifo rmization: for any p oint x ∈ M there exist functions u 1 , ..., u n ∈ O ( M ), forming lo cal co or dinates of the manifold M in a neig h bo rho od of x ; 3) holomorph ic c onvexity: for any compact set K ⊆ M its holomorph ic al ly c onvex hul l , i.e. a the set ˆ K = { x ∈ M : ∀ u ∈ O ( M ) | u ( x ) | 6 max y ∈ K | u ( y ) |} is a compact set in M . The complex space C n and v arious domains of holomor ph y in C n are examples o f Stein manifolds. A simplest example of a complex manifold which is not a Stein manifold is c omplex torus , i.e. the quotient group of the complex plane C ov er the lattice Z + i Z . (b) Outer en v elop es on M a nd rectangles in O ( M ) Op erations  and  . Here we shall sp end so me time o n studying rea l functions f on a ma nifold M , bo unded by 1 from b elow, f > 1 , i.e. f having range in the int erv al [1 , + ∞ ). Certainly , w e shall use record f : M → [1; + ∞ ) for such functions. As usual, we call a function f : M → [1 ; + ∞ ) lo c al ly b ounde d , if for each p oin t x ∈ M one can ﬁnd a neighbo rhoo d U ∋ x such that sup y ∈ U | f ( y ) | < ∞ Since f is b ounded by 1 from b elow, this condition is equiv a len t to the condition sup y ∈ U f ( y ) < ∞ Prop osition 2.1. F or e ach lo c al ly b ounde d funct ion f : M → [1 ; + ∞ ) the formula f  := { u ∈ O ( M ) : ∀ x ∈ M | u ( x ) | 6 f ( x ) } (2.1) deﬁnes an absolutely c onvex s et of functions f  ⊆ O ( M ) , c ontaining the identity function: 1 ∈ f  . Pr o of. The se t f  is compact since it is closed and b ounded in the Montel space O ( M ). Prop osition 2.2. F or any b ounde d set of fun ct ions D ⊆ O ( M ) c ontaining the identity function, 1 ∈ D , the formula D  ( x ) := sup u ∈ D | u ( x ) | , x ∈ M (2.2) deﬁnes a c ontinuous re al fun ction D  : M → R b ounde d by 1 fr om b elow: D  > 1 . § 2. STEIN MANIFOLDS: RECT ANGLES IN O ( M ) AND RHOMBUSES IN O ⋆ ( M ) 39 Pr o of. Note fro m the very b eginning that D ca n b e considered as co mpact. Consider for this its clos ure D . Since D is b ounded in a Mont el space O ( M ), D is compac t in O ( M ). The mapping u 7→ δ x ( u ) = u ( x ) is contin uous, so the image of the clo sure δ x ( D ) is contained in the clo sure of imag e δ x ( D ) δ x ( D ) ⊆ δ x ( D ) , As a coro llary , we obtain the following chain o f inequalities: sup λ ∈ δ x ( D ) | λ | 6 sup λ ∈ δ x ( D ) | λ | 6 sup λ ∈ δ x ( D ) | λ | = sup λ ∈ δ x ( D ) | λ | Therefore, sup λ ∈ δ x ( D ) | λ | = sup λ ∈ δ x ( D ) | λ | and the functions D  and D  coincide: D  ( x ) = sup u ∈ D | δ x ( u ) | = sup λ ∈ δ x ( D ) | λ | = sup λ ∈ δ x ( D ) | λ | = sup u ∈ D | δ x ( u ) | = D  ( x ) Thu s it is suﬃcient to c onsider the ca se when D is compac t. Let us take an a rbitrary co mpact set K ⊆ M a nd co nsider the space C ( K ) of contin uous functions on K (with the usual top ology of uniform conv ergence on K ). The restriction mapping u ∈ O ( M ) 7→ u | K ∈ C ( K ) is contin uous fro m O ( M ) into C ( K ), so the image D | K of a compact D in O ( M ) m ust b e compact in C ( K ). Hence, by the Arcela theorem D | K , is p oin twise b ounded and equicontin uous on K . Therefore the function D  ( x ) := sup u ∈ D | u ( x ) | , x ∈ K is contin uous on K . This is tr ue for every compa ct set K in M , so we obtain that D  is contin uous on M . Prop erties o f op erations  and  : f 6 g = ⇒ f  ⊆ g  , D ⊆ E = ⇒ D  6 E  (2.3) ( f  )  6 f , D ⊆ ( D  )  (2.4) (( f  )  )  = f  , (( D  )  )  = D  (2.5) Pr o of. Prop erties (2.3) a nd (2.4) a re evident, a nd (2.5) follows fro m them:  ( f  )  6 f = ⇒ (apply the o pera tion  ) = ⇒ (( f  )  )  ⊆ f  D ⊆ ( D  )  = ⇒ (substitution: D = f  ) = ⇒ f  ⊆ (( f  )  )   = ⇒ (( f  )  )  = f   D ⊆ ( D  )  = ⇒ (apply the op eration  ) = ⇒ D  6 (( D  )  )  ( f  )  6 f = ⇒ (substitution: f = D  ) = ⇒ (( D  )  )  6 D   = ⇒ (( D  )  )  = D  40 Outer en v elop es on M . Let us introduce the fo llo wing notations: f  := ( f  )  D  := ( D  )  (2.6) Then (2.3), (2.4), a nd (2.5) imply f  6 f , D ⊆ D  (2.7) f 6 g = ⇒ f  6 g  , D ⊆ E = ⇒ D  ⊆ E  (2.8) ( f  )  = f  ( D  )  = D  (2.9) Let us call a lo cally bo unded function g : M → [1 , + ∞ ), — an outer envelop e for a b ounde d set D ⊆ O ( M ), 1 ∈ D , if g = D  — an outer envelop e for a lo c al ly b ounde d fun ction f : M → [1; + ∞ ), if g = f  — an outer envelop e on M , if it s atisﬁes the following equiv a len t conditions: (i) g is an outer env elop e for some b ounded set D ⊆ O ( M ), 1 ∈ D , g = D  (ii) g is an outer env elop e for some lo cally b ounded function f : M → [1 ; + ∞ ), g = f  (iii) g is an outer env elop e for some for itself: g  = g Pr o of. The equiv alence of conditio ns (i), (ii), (iii) requires so me comments. ( i ) = ⇒ ( ii ). If g is an outer env elop e for so me b ounded set D , i.e. g = D  , then g = D  = (2.5) = (( D  )  )  = ( D  )  , i.e. g is an outer envelope for the function f = D  . ( ii ) = ⇒ ( iii ). If g is an outer envelope for some function f , i.e . g = f  , then g  = ( f  )  = (2.9) = f  = g , i.e. g is an outer envelope for itself. ( iii ) = ⇒ ( i ). If g is an outer env elop e for itself, i.e. g = g  = ( g  )  , then we put D = g  , and after that g be comes a n outer env elop e for D : g = D  . Prop erties o f outer env elop es : (i) Every outer envelop e g on M is a c ontinuous (p ositive) function on M . (ii) F or any lo c al ly b ounde d funct ion f : M → [1; + ∞ ) its outer envelop e f  is the gr e atest outer envelop e on M , majorize d by f : (a) f  is an outer envelop e on M , majorize d by f : f  6 f (b) if g is another out er envelop e on M , majorize d by f , g 6 f then g is majorize d by f  : g 6 f  § 2. STEIN MANIFOLDS: RECT ANGLES IN O ( M ) AND RHOMBUSES IN O ⋆ ( M ) 41 Rectangles in O ( M ) . W e call a set E ⊆ O ( M ), 1 ∈ E , — a r e ct angle, gener ate d by a lo c al ly b oun de d funct ion f : M → [1 ; + ∞ ), if E = f  — a r e ct angle, gener ate d by a b ounde d set D ⊆ O ( M ), 1 ∈ D , if E = D  — a r e ct angle in O ( M ), if the following equiv alent conditions hold: (i) E is a rectangle, generated by some lo cally b ounded function f : M → [1 ; + ∞ ) E = f  (ii) E is a rectangle, generated by some b ounded set D ⊆ O ( M ), 1 ∈ D , E = D  (iii) E is a rectangle, generated by itself: E = E  Pr o of. The eq uiv alence of conditions (i), (ii), (iii) is proved in the same wa y a s in the ca se o f o uter env elop es. Prop erties o f rectangles: (i) Every r e ctangle E in O ( M ) is an absolutely c onvex c omp act s et in O ( M ) . (ii) F or any b ounde d set D ⊆ O ( M ) the r e ctangle D  is t he smal lest r e ctangle in O ( M ) , c ontaining D : (a) D  is a r e ctangle in O ( M ) , c ont aining D : D ⊆ D  (b) if E is another r e ctangle in O ( M ) , c ontaining D , D ⊆ E , then E c ontains D  : D  ⊆ E (iii) The r e ctangles in O ( M ) form a fundamental system of c omp act sets in O ( M ) : every c omp act set D in O ( M ) is c ontaine d in s ome re ctangle. Theorem 2.1. The formulas D = f  , f = D  (2.10) establish a bije ction b etwe en outer en velop es f on M and r e ct angles D in O ( M ) . Pr o of. By deﬁnition of outer env elop es and recta ngles, the op erations f 7→ f  and D 7→ D  turn outer env elop es into re ctangles and rectangles into o uter env elop es. Moreover, these o pera tions are mutually inv erse on tho se tw o classes – if f is an outer env elop e, then f  = f , i.e. the comp osition of the oper ations  and  gives back to the initial function: f 7→ f  7→ f  = f Just like this the comp osition of the op eratio ns  and  returns to the se t D , if initially it was chosen a s a rectang le: D 7→ D  7→ D  = D 42 (c) Lemma on p olars Recall that p olar of a s et A in a lo cally c on vex spa ce X is the set A ◦ of linear contin uous functionals f : X → C , b ounded by 1 o n A : A ◦ = { f ∈ X ⋆ : sup x ∈ A | f ( x ) | 6 1 } If X is a stereo t yp e space and A a subset in the dual space X ⋆ , then b ecause o f the equality ( X ⋆ ) ⋆ = X it is conv enient to consider the p olar A ◦ ⊆ ( X ⋆ ) ⋆ as a subse t in X . W e denote this set by ◦ A (and again call it p olar of A ): ◦ A = { x ∈ X : sup f ∈ A | f ( x ) | 6 1 } An imp ortant obser v ation for us is that if A ⊆ X ⋆ , and we take its p olar ◦ A , and a fter that “the p olar of the pola r” ( ◦ A ) ◦ – this s et is ca lled bip olar of A – then it tur ns o ut that ( ◦ A ) ◦ is exactly the closed absolutely co n vex hull of the s et A in X ⋆ (i.e. the clos ure of the set of linea r combinations of the form P n i =1 λ i · a i , where a i ∈ A , P n i =1 | λ i | 6 1): ( ◦ A ) ◦ = absconv A (2.11) That is the essence the cla ssical the or em on bip olar a s applied to s tereot yp e spaces . In the sp ecial case when A = D is a set in O ( M ), its p olar D ◦ is the set o f analytical functiona ls α ∈ O ⋆ ( M ), b ounded by 1 on D : D ◦ = { α ∈ O ⋆ ( M ) : sup u ∈ D | α ( u ) | 6 1 } On the contrary , if A is a set of analytical functionals, A ⊆ O ⋆ ( M ), then its po lar ◦ A in O ( M ) is the set of functions u ∈ O ( M ) o n which a ll the functionals α ∈ A a re b ounded by 1: ◦ A = { u ∈ O ( M ) : sup α ∈ A | α ( u ) | 6 1 } Lemma 2.1 ( on p olars). The p assage to p olars p ossesses the fol lowing pr op erties. (a) F or every b ounde d set D in O ( M ) c ontaining the u n it, its out er envelop e D  is c onne cte d with its p olar D ◦ thr ough the identity 1 D  ( x ) = max { λ > 0 : λ · δ x ∈ D ◦ } (2.12) (b) F or any lo c al ly b ounde d funct ion f : M → [1 ; + ∞ ) the re ctangle f  is the p olar of functionals 1 f ( x ) · δ x : f  = ◦  1 f ( x ) · δ x ; x ∈ M  (2.13) (c) The p olar of the r e ctangle f  is an absolutely c onvex hul l of fun ctionals 1 f ( x ) · δ x : ( f  ) ◦ = absconv  1 f ( x ) · δ x ; x ∈ M  (2.14) Pr o of. (a) F or λ > 0 we hav e: λ · δ x ∈ D ◦ ⇐ ⇒ sup u ∈ D | λ · δ x ( u ) | 6 1 ⇐ ⇒ D  ( x ) = sup u ∈ D | δ x ( u ) | 6 1 λ ⇐ ⇒ λ 6 1 D  ( x ) § 2. STEIN MANIFOLDS: RECT ANGLES IN O ( M ) AND RHOMBUSES IN O ⋆ ( M ) 43 (b) is a refor m ulation of the deﬁnition of f  : u ∈ f  (2.1) ⇐ ⇒ ∀ x ∈ M | u ( x ) | = | δ x ( u ) | 6 f ( x ) ⇐ ⇒ ⇐ ⇒ sup x ∈ M     1 f ( x ) · δ x ( u )     6 1 ⇐ ⇒ u ∈ ◦  1 f ( x ) · δ x ; x ∈ M  (c) follows from (b) and from the theorem on bip olar: ( f  ) ◦ =  ◦  1 f ( x ) · δ x ; x ∈ M  ◦ = (2.11) = absconv  1 f ( x ) · δ x ; x ∈ M  (d) Inner en v elopes on M and rhom buses in O ⋆ ( M ) In this subsectio n we shall study closed absolutely conv ex neighbor hoo ds of z ero ∆ in O ⋆ ( M ), satisfying the following tw o equiv alen t conditions: (A) the p olar ◦ ∆ of ∆ c ontains the unit 1 ∈ O ( M ) : 1 ∈ ◦ ∆ (2.15) (B) the value of al l functionals α ∈ ∆ on the u n it 1 ∈ O ( M ) do es not exc e e d 1: ∀ α ∈ ∆ | α (1) | 6 1 (2.16) These conditions imply a nother one (which is not, how ever, eq uiv alent to (A) and (B)): (C) a functional of the form λ · δ x , wher e λ > 0 , c an b elong to ∆ only if λ 6 1 : ∀ x ∈ M ∀ λ > 0  λ · δ x ∈ ∆ = ⇒ λ 6 1  . (2.17) Pr o of. If λ · δ x ∈ ∆ , then ∀ u ∈ ◦ ∆ | λ · δ x ( u ) | 6 1. In particular, for u = 1 we hav e | λ · δ x (1) | = λ · 1 6 1 , i.e. λ 6 1. W e say that a function ϕ : M → (0 , + ∞ ) is lo c al ly sep ar ate d fr om zer o , if for each po in t x ∈ M there exists a neighbo rhoo d U ∋ x such that inf y ∈ U ϕ ( y ) > 0 Op erations  and ♦ . Prop osition 2.3. If a function ϕ : M → (0 , 1 ] is lo c al ly sep ar ate d fr om zer o, t hen the formula ϕ  := absconv { ϕ ( x ) · δ x ; x ∈ M } (2.18) deﬁnes an absolutely c onvex neighb orho o d of zer o in the sp ac e of analytic al functionals O ⋆ ( M ) , satisfying (2.15) - (2.1 7 ) . Pr o of. The function f ( x ) = 1 ϕ ( x ) is lo cally b ounded and has v alues in [1; + ∞ ). Hence, by Lemma 2.1, ( f  ) ◦ = (2.14) = absconv  1 f ( x ) · δ x ; x ∈ M  = absconv { ϕ ( x ) · δ x ; x ∈ M } This se t is a clo sed a bsolutely convex neighborho o d of zero in O ⋆ ( M ), since it is the p olar of the com- pact set f  in O ( M ). B esides this, since f > 1, the compact set f  contains the unit, so its p olar absconv { ϕ ( x ) · δ x ; x ∈ M } s atisﬁes co nditions (A),(B),(C) o n pa ge 43. 44 Prop osition 2.4. F or any close d absolutely c onvex neighb orho o d of zer o ∆ in t he sp ac e of analytic al functionals O ⋆ ( M ) , satisfying (2.15) - (2.1 6) the formula ∆ ♦ ( x ) := sup { λ > 0 : λ · δ x ∈ ∆ } (2.19) deﬁnes a c ontinuous and lo c al ly sep ar ate d fr om zer o function ∆ ♦ : M → (0; 1] . Pr o of. Since ∆ is a neighborho o d of z ero in O ⋆ ( M ), its p olar D = ◦ ∆ is a compac t in O ⋆ ( M ), and from (2.15) we have 1 ∈ D . By P ropo sition 2 .2, the outer env elop e D  of this compact set is contin uous and lo cally b ounded. Hence, the function ∆ ♦ ( x ) := sup { λ > 0 : λ · δ x ∈ ∆ = D ◦ } = (2.12) = 1 D  ( x ) is contin uous a nd lo cally sepa rated from zero . The following prop erties are prov ed simila rly with (2 .3), (2.4) and (2.5). Prop erties o f the op erations  and ♦ : ϕ 6 ψ = ⇒ ϕ  ⊆ ψ  , ∆ ⊆ Γ = ⇒ ∆ ♦ 6 Γ ♦ (2.20) ϕ 6 ( ϕ  ) ♦ , ( ∆ ♦ )  ⊆ ∆ (2.21) (( ϕ  ) ♦ )  = ϕ  , (( ∆ ♦ )  ) ♦ = ∆ ♦ (2.22) Inner env elo pe s o n M . Let us use the following supplementary nota tions: ϕ  ♦ := ( ϕ  ) ♦ ∆ ♦  := ( ∆ ♦ )  (2.23) F o rm ulas (2.3), (2.4), (2.5) imply ϕ 6 ψ = ⇒ ϕ  ♦ 6 ψ  ♦ , ∆ ⊆ Γ = ⇒ ∆ ♦  ⊆ Γ ♦  (2.24) ϕ 6 ϕ  ♦ , ∆ ♦  ⊆ ∆ (2.25) ( ϕ  ♦ )  ♦ = ϕ  ♦ ( ∆ ♦  ) ♦  = ∆ ♦  (2.26) Let us call a lo cally separated from zero function ψ : M → (0 ; 1] — an inner envelop e for an absolutely c onvex neighb orho o d of zer o ∆ in O ⋆ ( M ), 1 ∈ ◦ ∆ , if ψ = ∆ ♦ — an inner envelop e for a lo c al ly sep ar ate d fr om zer o funct ion ϕ : M → (0; 1], if ψ = ϕ  ♦ — an inner envelop e on M , if it s atisﬁes the fo llo wing equiv alent co nditions: (i) ψ is an inner env elop e for some a bsolutely conv ex neighbor hoo d of zero ∆ in O ⋆ ( M ), 1 ∈ ◦ ∆ , ψ = ∆ ♦ § 2. STEIN MANIFOLDS: RECT ANGLES IN O ( M ) AND RHOMBUSES IN O ⋆ ( M ) 45 (ii) ψ is an inner env elop e for so me lo cally separa ted from zero function ϕ : M → (0; 1]. ψ = ϕ  ♦ (iii) ψ is an inner env elop e for itself: ψ  ♦ = ψ The equiv alence of the conditions (i), (ii), (iii) is proved in the same wa y as for outer envelopes. Prop erties o f the i nner en v elop es : (i) Every inner envelop e ψ on M is a c ontinuous function on M . (ii) F or any lo c al ly sep ar ate d fr om zer o funct ion ϕ : M → (0; 1 ] its inner envelop e ϕ  ♦ is the le ast inner envelop e on M , majorizing ϕ : (a) ϕ  ♦ is an inner en velop e on M , majorizing ϕ : ϕ 6 ϕ  ♦ (b) if ψ is another inner en velop e on M , majorizing ϕ , ϕ 6 ψ then ψ majorizes ϕ  ♦ : ϕ  ♦ 6 ψ Rhombuses in O ⋆ ( M ) W e call a set Γ ⊆ O ⋆ ( M ), — a rhombus, gener ate d by a lo c al ly sep ar ate d fr om zer o function ϕ : M → (0 ; 1], if Γ = ϕ  — a rhombus, gener ate d by an absolutely c onvex neighb orho o d of zer o ∆ ⊆ O ⋆ ( M ), 1 ∈ ◦ ∆ , if Γ = ∆ ♦  — a rhombus in O ⋆ ( M ), if it sa tisﬁes the following eq uiv alent conditions: (i) Γ is a rhombus, generated by some lo cally separ ated from zer o function ϕ : M → (0; 1], Γ = ϕ  (ii) Γ is a rho m bus, generated by some absolutely convex neighborho o d of z ero ∆ ⊆ O ⋆ ( M ), 1 ∈ ◦ ∆ , Γ = ∆ ♦  (iii) the r hom bus g enerated by Γ , co incides with Γ : Γ = Γ ♦  Pr o of. The equiv alence of conditions (i), (ii), (iii) is prov ed in the same way as in the case o f rectangle s. Prop erties o f rhombuses: (i) Every rhombus ∆ in O ⋆ ( M ) is a close d absolutely c onvex neighb orho o d of zer o in O ⋆ ( M ) . 46 (ii) F or any neighb orho o d of zer o ∆ ⊆ O ⋆ ( M ) the rhombus ∆ ♦  is t he gr e atest rhombus in O ⋆ ( M ) , c ont aine d in ∆ : (a) ∆ ♦  is a rhombus in O ⋆ ( M ) , c ontaine d in ∆ : ∆ ♦  ⊆ ∆ (b) if Γ is another rhombus in O ⋆ ( M ) , c ontaine d in ∆ , Γ ⊆ ∆, then Γ is c ontaine d in ∆ ♦  : Γ ⊆ ∆ ♦  (iii) Rhombuses in O ⋆ ( M ) form a fundamental system of neighb orho o ds of zer o in O ⋆ ( M ) : every neigh- b orho o d of zer o Γ in O ⋆ ( M ) c ont ains some rhombus. By analogy with Theore m 2.1 one can prove Theorem 2.2. The formulas ∆ = ϕ  , ϕ = ∆ ♦ (2.27) establish a bije ction b etwe en inner envelop es ϕ on M and rhombuses ∆ in O ⋆ ( M ) . (e) Dualit y b et w een rectangles and rhom buses Lemma 2.1 implies Theorem 2.3. The fol lowing e qualities hold ( f  ) ◦ =  1 f   , ◦ ( ϕ  ) =  1 ϕ   , (2.28) ( D ◦ ) ♦ = 1 D  , ( ◦ ∆ )  = 1 ∆ ♦ , (2.29) 1 f  =  1 f   ♦ , 1 ϕ  ♦ =  1 ϕ   , (2.30)  D   ◦ =  D ◦  ♦  , ◦  ∆ ♦   =  ◦ ∆   , (2.31) wher e f : M → [1; + ∞ ) is an arbitr ary lo c al ly b ounde d function, ϕ : M → (0; 1] an arbitr ary lo c al ly sep ar ate d fr om zer o function, D an arbitr ary absolutely c onvex c omp act set in O ( M ) , ∆ an arbitr ary close d absolutely c onvex neighb orho o d of zer o in O ⋆ ( M ) . Pr o of. 1. The ﬁr st formula in (2.28) follows from (2.14): ( f  ) ◦ = (2.14) = absco nv  1 f ( x ) · δ x ; x ∈ M  = (2.18) =  1 f   After that the substitution f = 1 ϕ gives the second formula:  1 ϕ   ◦ = ϕ  = ⇒  1 ϕ   = ◦  1 ϕ   ◦ ! = ◦  ϕ   § 2. STEIN MANIFOLDS: RECT ANGLES IN O ( M ) AND RHOMBUSES IN O ⋆ ( M ) 47 2. The ﬁrst formula in (2.2 9) follows from (2 .12): 1 D  ( x ) = (2.12) = max { λ > 0 : λ · δ x ∈ D ◦ } = (2.19) = ( D ◦ ) ♦ ( x ) = ⇒ ( D ◦ ) ♦ = 1 D  Then the substitution D = ◦ ∆ gives the second formula: D  = 1 ( D ◦ ) ♦ = ⇒ ( ◦ ∆ )  = 1 (( ◦ ∆ ) ◦ ) ♦ = 1 ∆ ♦ 3. Now the ﬁrst formula in (2.30) follows from the ﬁrst formula in (2 .29) and from the ﬁrst formula in (2.28) by substitution D = f  : 1 D  = ( D ◦ ) ♦ = ⇒ 1 f  = ( f  ◦ ) ♦ = (2.28) =  1 f  ♦ The second formula in (2.30) follows fro m the seco nd formula in (2.29) and the second fo rm ula in (2.28) after the substitution ∆ = ϕ  : 1 ∆ ♦ = ( ◦ ∆ )  = ⇒ 1 ϕ ♦ =  ◦ ( ϕ  )   = (2.28) =  1 ϕ   4. The ﬁrs t formula in (2.3 1) follows from the ﬁr st formula in (2 .29) a nd from the ﬁrst fo rm ula in (2.28): ( D ◦ ) ♦ = 1 D  = ⇒ ( D ◦ ) ♦ =  1 D    = (2.28) = ( D  ) ◦ Finally the s econd formula in (2.31) follows from the seco nd for m ula in (2.29) and from the second f ormula in (2.28): ( ◦ ∆ )  = 1 ∆ ♦ = ⇒ ( ◦ ∆ )  =  1 ∆ ♦   = (2.28) = ◦  ∆ ♦   Theorem 2.3 implies tw o impor tan t prop ositions. Theorem 2.4. The p assage to the inverse funct ion f = 1 ϕ , ϕ = 1 f (2.32) establish a bije ction b etwe en the outer envelop es f and the inner envelop es ϕ on M . Pr o of. If ϕ is an inner en velope, then ϕ ♦ = ϕ , hence  1 ϕ   = (2 .30) = 1 ϕ  ♦ = 1 ϕ , i.e. 1 ϕ is an outer env elop e. On the co n trary , if f is an outer e n velope, then f  = f , therefor e  1 f   ♦ = (2.30) = 1 f   = 1 f , i.e. 1 f is an inner env elop e. Theorem 2.5. The p assage to p olar D = ◦ ∆, ∆ = D ◦ (2.33) establishes a bije ction b etwe en r e ctangles D in O ( M ) and rhombuses ∆ in O ⋆ ( M ) . Pr o of. If ∆ is a rhombus, then ∆ ♦  = ∆ , hence ( ◦ ∆ )  = (2 .31) = ◦ ( ∆ ♦  ) = ◦ ∆ , i.e. ◦ ∆ is a rectangle. On the contrary , if D is a rectangle, then D  = D , hence ( D ◦ ) ♦  = (2.31) = ( D  ) ◦ = D ◦ , i.e. D ◦ is a rhombus. Since the pa ssage to p olar is a bijection b etw een closed absolutely co n vex sets, it is a bijection betw een r ectangles and rho m buses. 48 Theorems 2.4, 2.5 and 2.3 imply Theorem 2.6. The fol lowing e qualities hold:  ( f  ) ◦  ♦ = 1 f ,  ◦ ( ϕ  )   = 1 ϕ (2.34)  1 ( D ◦ ) ♦   = D , 1 ( ◦ ∆ )  !  = ∆. (2.35) wher e f : M → [1; + ∞ ) is an arbitr ary outer envelop e, ϕ : M → (0; 1 ] an arbitr ary inn er envelop e, D an arbitr ary r e ctangle in O ( M ) , ∆ an arbitr ary rhombus in O ⋆ ( M ) . Pr o of. If f is an o uter env elop e, then by Theorem 2.4, 1 f is an inner env elop e, so  ( f  ) ◦  ♦ = (2.28) =  1 f   ♦ = 1 f The other formulas ar e proved by ana logy . § 3 Stein group s and Hopf algebras connected to them (a) Stein groups, linear groups and algebraic groups A complex Lie group G is called a Stein gr oup , if G is a Stein manifold [14 ]. By the Matsushima-Mor imoto Theorem [24, XI I I.5.9], for complex gro ups this is equiv alent to the condition of holo morphic separability we mentioned at the page 3 8: ∀ x 6 = y ∈ G ∃ u ∈ O ( G ) u ( x ) 6 = u ( y ) Dimension of a Stein gr oup is its dimensio n a s a complex manifold. Spec ial cases of Stein groups are line ar c omplex gr oups . They are deﬁned as complex Lie gro ups which can be r epresent ed a s clo sed complex Lie subgroups in the g eneral linear gro up GL n ( C ). In o ther words, a complex gro up G is linear , if it is is omorphic to so me clos ed complex subgro up H in GL n ( C ) (i.e. there is an isomor phism of gro ups ϕ : G → H which is a t the same time a biholomor phic mapping). Even more nar row cla ss are c omplex aﬃne algebr aic gr oups . These a re subgroups H in GL n ( C ), which are at the s ame time algebraic submanifolds. This means that the group H must b e a common set o f zero es fo r some ﬁnite set of p olynomials u 1 , ..., u k on GL n ( C ) (by p olynomial here we ca n understand a po lynomial of matrix elements): H = { x ∈ GL n ( C ) : u 1 ( x ) = ... = u k ( x ) = 0 } If a complex gro up G is isomorphic to some algebr aic group H (i.e. there exists an isomo rphism of groups ϕ : G → H which is at the sa me time a biholomo rphic mapping ), then G is also co nsidered a n a lgebraic group, since the a lgebraic opera tions on G are regular mappings with res pect to the structure of alg ebraic manifold inherited fro m H . A Stein group G is called c omp actly gener ate d , if it ha s a gener ating co mpact set, i.e. a compact set K ⊆ G s uc h that G = [ n ∈ N K n , K n = K · ... · K | {z } n factors Let us note some exa mples. Example 3.1. Complex torus we hav e men tioned in § 2, i.e. the quotient group C / ( Z + i Z ), is an example of a complex gro up, which is not a Stein group. § 3. STEIN GROUPS AND HO PF ALGEBRAS CO NNECTED TO THEM 49 Example 3.2. Every discr ete gr oup G is a Stein gr oup (of zero dimension). Compact se ts in G a re nothing more tha n ﬁnite sets, hence G will b e compa ctly gener ated if a nd only if it is ﬁnitely generated. Thu s, say , a free group with inﬁnite set of genera tors can be cons idered as an example a Stein gr oup which is not compa ctly gener ated. Example 3. 3 . A discrete gr oup G is a lgebraic if and only if it is ﬁ nite . In this cas e it can b e represented as a gr oup of tra nsformations of the space C n , whe re n = card G is the num b er of e lemen ts of G . F or this C n m ust b e repre sen ted as the s pace o f functions from G into C : x ∈ C G ⇐ ⇒ x : G → C , Then the imbedding o f G in to GL ( C G ) (the gr oup of a no ndegenerate linear tra nsformations o f the spa ce C G ) is deﬁned by the for m ula ϕ : G → GL ( C G ) : ϕ ( g )( x )( h ) = x ( h · g ) , g , h ∈ G, x ∈ C G Example 3.4. The general linear group GL n ( C ), i.e. the group of a nondegenerate linear transformations of the s pace C n , is an algebra ic group (of dimension n 2 ). Certa inly , GL n ( C ) is co mpactly g enerated. As a cor ollary , every line ar gr oup is c omp actly gener ate d . Example 3. 5 . The additive group of co mplex num b ers C is a complex algebra ic group (of dimension 1 ), since it can b e embedded into GL (2 , C ) by for m ula ϕ ( λ ) =  1 λ 0 1  , λ ∈ C Example 3.6. The additive g roup Z of integers is a complex linear group (of dimensio n 0 ), since it can be embedded into GL (2 , C ) by the s ame for m ula ϕ ( n ) =  1 n 0 1  , n ∈ Z But the diﬀerence with C is that Z is no t alg ebraic: neither in this embedding into GL n ( C ), nor in a n y other one, Z is a common set for a family of po lynomials (this is a result of the fact that Z is discr ete and inﬁnite). Example 3.7 . A multiplicativ e gro up C × of nonzero complex num ber s C × = C \ { 0 } can b e represe n ted as gener al linear gr oup (of nondegener ate transforma tions of the spa ce C ), C × ∼ = GL 1 ( C ) , Thu s C × is a complex alge braic gro up (of dimension 1). W e call this gr oup c omplex cir cle . Example 3.8 . Cons ider the action of g roup C on itself by exp onen ts: ϕ : C → Aut ( C ) , ϕ ( a )( x ) = x · e a The semidirect pro duct of C and C with resp ect to this actio n, i.e. a gro up C ⋉ C , coinciding with the Cartesian pro duct C × C , but endow ed with a more complicated multiplication ( a, x ) · ( b, y ) := ( a + b , x · e b + y ) is a (connected) linear c omplex gr oup, since it ca n b e embedded into GL 3 ( C ) by the homo morphism ( x, a ) 7→   e a 0 0 x 1 0 0 0 e ia   But C ⋉ C is not algebra ic, since its center Z ( C ⋉ C ) = { (2 π in, 0); n ∈ Z } is an inﬁnite disc rete subg roup (this do es not happ en with algebr aic gro ups). 50 (b) Hopf algebras O ( G ) , O ⋆ ( G ) , R ( G ) , R ⋆ ( G ) The Hopf a lgebras C G and C G we were talking a bout in § 1(c), are int eresting not so in themselves but more a s guiding examples for v arious similar constructio ns ar ising in situations whe n the initial gro up G is endow ed with some supplementary struc ture and functions on G preserve this structure. In particula r, in [1 , exa mples 10.24 -10.27] the author noted standa rd examples of a lgebras of functions a nd functionals, which ar e stereo t yp e Hopf a lgebras. If we add Ho pf a lgebras C G and C G to those ex amples, we obtain the following list: class of groups algebra o f functions algebra o f functiona ls ”pure” gro ups algebra C G of all functions on G algebra C G of p oin t charges on G algebraic gro ups algebra R ( G ) of p olynomials on G algebra R ⋆ ( G ) of currents of degr ee 0 on G Stein groups algebra O ( G ) of holomor phic functions on G algebra O ⋆ ( G ) of analytica l functionals on G Lie gro ups algebra E ( G ) of smo oth functions on G algebra E ⋆ ( G ) of distributions on G lo cally c ompact gr oups algebra C ( G ) of contin uous functions on G algebra C ⋆ ( G ) of Radon measure s on G In [1] the las t four ex amples were men tioned without pr oo f, so we see ﬁt to expla in her e why thos e algebras a re indeed stereo t yp e Hopf algebr as. Stein gro ups and algebra ic groups will b e case studies for us. Hopf algebras O ( G ) and O ⋆ ( G ) on a Stein group G . If G is a Stein g roup, then the prop osition that alg ebra O ( G ) of holomo rphic functions on G (with the usual top ology of uniform convergence o n compact sets in G ) is a n injective stereotype Hopf alg ebra, is proved exactly like this was done fo r C G . A signiﬁcant asp ect in tho se rea sonings is the isomor phism o f functors connecting Car tesian pro duct of groups × with the co rresp onding tensor pro ducts of functional spa ces ⊙ and ⊛ [1, Theorem 8.13 ], O ( G × H ) ρ G,H ∼ = O ( G ) ⊙ O ( H ) @ − 1 ∼ = O ( G ) ⊛ O ( H ) (3.1) – here ρ G,H is deﬁned by identit y analog ous to (1.25): ρ G,H ( u ⊡ v ) = u ⊙ v , u ∈ O ( G ) , v ∈ O ( H ) (3.2) and the function u ⊡ v is ag ain deﬁned by formula (1.24). After deﬁning ρ G,H , the multiplication and comultiplication in O ( G ) a re deﬁned initially on (or with v alues in) the spa ce O ( G × G ) of functions o n the Cartesian square G × G , and then the passag e to tensor square is carr ied out with the help of the isomorphis m ρ G,G : m ultiplication: µ = e µ ◦ ρ G,G : O ( G ) ⊙ O ( G ) → O ( G ) , e µ ( v )( t ) = v ( t, t ) (3.3) unit: ι : C → O ( G ) , ι ( λ )( t ) = λ (3.4) comultiplication: κ = ρ G,G ◦ e κ : O ( G ) → O ( G ) ⊙ O ( G ) , e κ ( u )( s, t ) = u ( s · t ) (3.5) counit: ε : O ( G ) → C , ε ( u ) = u (1 G ) (3.6) antipo de: σ : O ( G ) → O ( G ) , σ ( u )( t ) = u ( t − 1 ) (3.7) § 3. STEIN GROUPS AND HO PF ALGEBRAS CO NNECTED TO THEM 51 This can b e illustrated by the following pictur e: O ( G × G ) C O ( G ) O ( G ) C O ( G ) ⊙ O ( G ) ' ' O O O O O O O e µ   ρ G,G / / ι 7 7 o o o o o o o e κ ' ' O O O O O O O κ / / ε 7 7 o o o o o o o µ (3.8) The fact that this deﬁnes a n injective stereotype Hopf algebr a is proved literally like Theorem 1.7. And again, like in the case of C G , the Hopf algebr a O ( G ) b ecomes a pro jective Hopf algebra (hence, a rig id Hopf algebra in the sens e of deﬁnition § 1(b)), b ecause of the second equality in (3.1) . Theorem 3.1. F or any Stein gr oup G — the algebr a O ( G ) of holomorphic functions on G is a rigid st er e otyp e Hopf algebr a with r esp e ct to algebr aic op er ations, deﬁne d by formulas (3.3) - (3.7) ; — its dual algebr a O ⋆ ( G ) of analytic al funct ional s on G is a rigid ster e otyp e Hopf algebr a with r esp e ct to the dual algebr aic op er ations. If in addition t he gr oup G is c omp actly gener ate d, then O ( G ) is a nu cle ar Hopf-F r´ echet algebr a, and O ⋆ ( G ) a nucle ar Hopf-Br auner algebr a. Hopf alg e bras R ( G ) and R ⋆ ( G ) on an aﬃne alge braic group G . Let G b e an aﬃne algebraic group, R ( G ) the algebra of p olynomials on G (with the strongest lo cally conv ex top olog y). Like in the previous case s, the identit y ρ G,H ( u ⊡ v ) = u ⊙ v , u ∈ R ( G ) , v ∈ R ( H ) (again, u ⊡ v is deﬁned by formula (1.24)) deﬁnes a n isomor phism of functor s, connecting the Cartesia n pro duct of g roups × and tensor pro ducts of functional spac es ⊙ and ⊛ [1, The orem 8.16 ]: R ( G × H ) ρ G,H ∼ = R ( G ) ⊙ R ( H ) @ − 1 ∼ = R ( G ) ⊛ R ( H ) (3.9) Those iso morphisms then deﬁne algebr aic op erations on R ( G ) by formulas, analo gous to (3.3)- (3.7). As a result we come to the following Theorem 3.2. F or any aﬃne algebr aic gr oup G — the algebr a R ( G ) of p olynoimals on G is a rigid st er e otyp e H opf-B r auner algebr a; — the dual algebr a R ⋆ ( G ) of cu r re nts of de gr e e 0 on G is an inje ctive ster e otyp e Hopf-F r ´ echet algebr a. Con v olutions i n R ⋆ ( G ) and O ⋆ ( G ) . F or some further calculations it is useful to recor d the for m ulas deﬁning co n volution in the algebr as of functionals R ⋆ ( G ) a nd O ⋆ ( G ). This deﬁnition is anticipated by formulas for shift, antipo de and convolution b etw een a functional and a function: ( u · a )( x ) := u ( a · x ) , ( a · u )( x ) := u ( x · a ) , u ∈ O ( G ) , a, x ∈ G (3.10) ( α · a )( u ) := α ( a · u ) , ( a · α )( u ) := α ( u · a ) , α ∈ O ⋆ ( G ) , u ∈ O ( G ) , a ∈ G (3.11) ˜ u ( x ) := u ( x − 1 ) , ˜ α ( u ) := α ( ˜ u ) , α ∈ O ⋆ ( G ) , u ∈ O ( G ) , x ∈ G (3.12) ( α ∗ u )( x ) := α ( g x · u ) , ( u ∗ α )( x ) := α ( g u · x ) α ∈ O ⋆ ( G ) , u ∈ O ( G ) , x ∈ G (3.13) Then the conv olution of functionals is deﬁned by fo rm ula α ∗ β ( u ) := α ( ] β ∗ ˜ u ) = ˜ α ( β ∗ ˜ u ) = β ( ˜ α ∗ u ) α, β ∈ O ⋆ ( G ) , u ∈ O ( G ) (3.14) In particular , the conv olution with the delta-functional is a shift: δ a ∗ β = a · β β ∗ δ a = β · a β ∈ O ⋆ ( G ) , a ∈ G (3.15) 52 (c) Examples It is us eful to illus trate the latter theor ems by several examples. F or this we choose the main exa mples of Ab elian Stein gr oups – Z , C × , C – a part fr om everything else, these ex amples will be useful b elow in § 6 and § 7. Algebras O ( Z ) and O ⋆ ( Z ) . W e have already noted in Exa mple 3.6 that the group Z of integers can b e considered as a co mplex group (of dimensio n 0). Since Z is discrete, each function on Z is automatically holomorphic, s o the alge bra O ( Z ) for mally coincides with the alg ebra C Z , and the algebr a O ⋆ ( Z ) with the algebra C Z : O ( Z ) = C Z , O ⋆ ( Z ) = C Z As a co rollary the structure of these a lgebras is describ ed by for m ulas (1.19)-(1.2 3 ): the c hara cteristic functions of single tons 1 n ( m ) = ( 0 , m = n 1 , m = n , m ∈ Z , n ∈ Z (3.16) form a basis in the stereotype space O ( Z ) = C Z , and delta-functionals δ k ( u ) = u ( k ) , u ∈ O ( Z ) a dual (algebr aic) basis in O ⋆ ( Z ) = C Z : h 1 n , δ k i = ( 0 , n 6 = k 1 , n = k , It is c on venien t to represent elements of O ( Z ) and O ⋆ ( Z ) in the form of series (which conv erge in thes e spaces) u ∈ O ( Z ) = C Z ⇐ ⇒ u = X n ∈ Z u ( n ) · 1 n , u ( n ) = δ n ( u ) , (3.17) α ∈ O ⋆ ( Z ) = C Z ⇐ ⇒ α = X n ∈ Z α n · δ n , α n = α (1 n ) , (3.18) where the action of α o n u is describ ed by formula h u, α i = X n ∈ Z u ( n ) · α n The op erations of multiplication in O ( Z ) = C Z and in O ⋆ ( Z ) = C Z are repres en ted by serie s: u · v = X n ∈ Z u ( n ) · v ( n ) · 1 n , α ∗ β = X k ∈ Z X i ∈ Z α i · β k − i ! · δ k , (3.19) (in the ﬁrs t cas e this is the co ordinate-wis e multiplication, a nd in the second case the multiplication of power serie s). Prop osition 3.1. The algebr a O ( Z ) = C Z of functions on Z is a nu cle ar Hopf-F r´ echet algebr a with t he top olo gy gener ate d by s eminorms || u || N = X | n | 6 N | u ( n ) | , N ∈ N . (3.20) § 3. STEIN GROUPS AND HO PF ALGEBRAS CO NNECTED TO THEM 53 and with algebr aic op er ations deﬁne d on b asis elements 1 k by formulas 1 m · 1 n = ( 1 m , m = n 0 , m 6 = n 1 R ⋆ ( C × ) = X n ∈ Z 1 n (3.21) κ (1 n ) = X m ∈ Z 1 m ⊙ 1 n − m ε (1 n ) = ( 1 , n = 0 0 , n 6 = 0 (3.22) σ (1 n ) = 1 − n (3.23) Prop osition 3. 2 . The algebr a O ⋆ ( Z ) = C Z of p oint char ges on Z is a n ucle ar Hopf-Br auner algebr a with the top olo gy gener ate d by seminorms ||| α ||| r = X n ∈ Z r n · | α n | ( r n > 0 ) (3.24) and algebr aic op er ations deﬁne d on b asis monomials δ k by formulas δ k ∗ δ l = δ k + l 1 R ( C × ) = δ 0 (3.25) κ ( δ k ) = δ k ⊛ δ k ε ( z k ) = 1 (3.26) σ ( δ k ) = δ − k (3.27) Pr o of. It ca n be not obvious here that semino rms (3.24) indeed deﬁne the top ology in O ⋆ ( Z ) = C Z . The auxiliary statement used here will b e als o useful for us b elow in Lemma 6.3, so we formulate it separately: Lemma 3.1. If p is a c ontinuous seminorm on O ⋆ ( Z ) , and r n = p ( δ n ) , then p is majorize d by seminorm (3.24) : p ( α ) 6 ||| α ||| r (3.28) Pr o of. p ( α ) = p X n ∈ Z α n · δ n ! 6 X n ∈ Z | α n | · p ( δ n ) = X n ∈ Z | α n | · r n = ||| α ||| r (3.29) Algebras R ( C × ) , R ⋆ ( C × ) , O ( C × ) , O ⋆ ( C × ) . In Example 3.7 we denoted by C × the multiplicativ e group of nonzero co mplex num be rs: C × := C \ { 0 } . (the multiplication in C × is the usual multiplication of co mplex num b ers). W e ca ll this g roup c omplex cir cle . The algebra R ( C × ) of p olynomials on C × consists of Laur en t polyno mials, i.e. of function of the form u = X n ∈ Z u n · z n where z n are mo nomials on C × : z n ( x ) := x n , x ∈ C × , n ∈ Z (3.30) and almost all u n ∈ C v a nish: card { n ∈ Z : u n 6 = 0 } < ∞ 54 And the algebra R ⋆ ( C × ) of currents on C × consists of functionals α = X k ∈ Z α k · ζ k where ζ k is the functional o f ﬁnding the k -th Laur en t co eﬃcient: ζ k ( u ) = 1 2 π i Z | z | =1 u ( z ) z k +1 d z = Z 1 0 e − 2 π ikt u ( e 2 π it ) d t, u ∈ R ( C × ) (3.31) and α k ∈ C is an ar bitrary sequence. Monomials ζ k and z n act on each other by for m ula h z n , ζ k i = ( 1 , n = k 0 , n 6 = k , (3.32) so the action of a cur ren t α on a p olynomial u is describ ed by formula h u, α i = X n ∈ Z u n · α n and the op erations of multiplication in R ( C × ) and in R ⋆ ( C × ) are represented by ser ies as follows: u · v = X n ∈ Z X i ∈ Z u i · v n − i ! · z n , α ∗ β = X k ∈ Z α k · β k · ζ k , (3.33) (in the ﬁrst ca se this is the multiplication of series, and in the second case the co ordinate-wise multipli- cation). Prop osition 3.3. The mapping u ∈ R ( C × ) 7→ { u k ; k ∈ Z } ∈ C Z is an isomorphism of nu cle ar Hopf-Br auner algebr as R ( C × ) ∼ = C Z (3.34) Prop osition 3. 4 . The algebr a R ( C × ) of p olynomial s on the c omplex cir cle C × is a nucle ar Hopf-F r ´ echet algebr a with the top olo gy gener ate d by seminorms ||| u ||| r = X n ∈ Z r n · | u n | ( r n > 0) (3 .35) and algebr aic op er ations deﬁne d on monomials z k by formulas z k · z l = z k + l 1 R ( C × ) = z 0 (3.36) κ ( z k ) = z k ⊙ z k ε ( z k ) = 1 (3.37) σ ( z k ) = z − k (3.38) Prop osition 3.5. The mapping α ∈ R ⋆ ( C × ) 7→ { α k ; k ∈ Z } ∈ C Z is an isomorphism of nu cle ar Hopf-F r´ echet algebr as R ⋆ ( C × ) ∼ = C Z (3.39) § 3. STEIN GROUPS AND HO PF ALGEBRAS CO NNECTED TO THEM 55 Prop osition 3. 6. The algebr a R ⋆ ( C × ) of curr ents on the c omplex cir cle C × is a nucle ar Hopf-F r´ echet algebr a with the top olo gy gener ate d by seminorms || α || N = X | n | 6 N | α n | ( N ∈ N ) (3.40) and algebr aic op er ations deﬁne d on b asis elements ζ k by formulas ζ k ∗ ζ l = ( ζ l , k = l 0 , k 6 = l 1 R ⋆ ( C × ) = X n ∈ Z ζ n (3.41) κ ( ζ k ) = X l ∈ Z ζ l ⊛ ζ k − l ε ( ζ k ) = ( 1 , k = 0 0 , k 6 = 0 (3.42) σ ( ζ k ) = ζ − k (3.43) As usual, by sym b ol O ( C × ) we denote the a lgebra of holomorphic functions on the co mplex cir cle C × , and by O ⋆ ( C × ) its dual algebr a of analytic functionals o n C × . It is useful to repr esen t the e lemen ts of algebra s O ( C × ) and O ⋆ ( C × ) by series u ∈ O ( C × ) ⇐ ⇒ u = X n ∈ Z u n · z n , u n ∈ C : ∀ C > 0 X n ∈ Z | u n | · C | n | < ∞  u n = ζ n ( u )  (3.44) α ∈ O ⋆ ( C × ) ⇐ ⇒ α = X n ∈ Z α n · ζ n , α n ∈ C : ∃ C > 0 ∀ n ∈ N | α n | 6 C | n |  α n = α ( z n )  (3.45) Like in the case o f R ( C × ) and R ⋆ ( C × ), the action of α o n u is describ ed by formula h u, α i = X n ∈ Z u n · α n , and the op erations of m ultiplication in O ( C × ) and O ⋆ ( C × ) ca n b e wr itten a s the usual mult iplication of series in the ﬁr st case, and a co ordinate-wise multiplication in the seco nd case: u · v = X n ∈ Z X i ∈ Z u i · v n − i ! · z n , α ∗ β = X n ∈ Z α n · β n · ζ n , (3.46 ) Prop osition 3. 7. The algebr a O ( C × ) of holo morphic functions on the c omplex cir cle C × is a nucle ar Hopf-F r ´ echet algebr a with t he t op olo gy gener ate d by seminorms || u || C = X n ∈ Z | u n | · C | n | , C > 1 . (3.47) and algebr aic op er ations deﬁne d on monomials z k by t he same formulas (3.36) - (3.38) as in the c ase of R ( C × ) : z k · z l = z k + l 1 O ( C × ) = z 0 κ ( z k ) = z k ⊙ z k ε ( z k ) = 1 σ ( z k ) = z − k Pr o of. Every usual seminorm | u | K = max x ∈ K | u ( x ) | , where K is a compact set in C × , is ma joriz ed by some seminorm || u || C , namely the one with C = max x ∈ K max {| x | , 1 | x | } : | u | K = max x ∈ K | u ( x ) | = max x ∈ K      X n ∈ Z u n · x n      6 max x ∈ K X n ∈ Z | u n | · | x n | 6 X n ∈ Z | u n | · C | n | = || u || C 56 On the contrary , for every num b er C > 1 we ca n take a compact set K = { t ∈ C : 1 C +1 6 | t | 6 C + 1 } , and then from the Cauchy formulas for Laurent co eﬃcients | u n | 6 | u | K · min  ( C + 1) n ; 1 ( C + 1) n  = | u | K · ( C + 1 ) −| n | = | u | K ( C + 1) | n | we hav e tha t || u || C is ma jorized by | u | K (with co eﬃcien t ( C + 1 ) 2 ): || u || C = X n ∈ Z | u n | · C | n | 6 X n ∈ Z | u | K ( C + 1) | n | · C | n | 6 | u | K · X n ∈ Z  C C + 1  | n | = ( C + 1) 2 · | u | K Prop osition 3. 8. The algebr a O ⋆ ( C × ) of analytic al functionals on t he c omplex cir cle C × is a nucle ar Hopf-Br auner algebr a with the top olo gy gener ate d by seminorms ||| α ||| r = sup u ∈ E r | α ( u ) | = X n ∈ Z r n · | α n | r n > 0 : ∀ C > 0 X n ∈ Z r n · C | n | < ∞ ! (3.48) and algebr aic op era tions deﬁne d on b asis elements ζ k by t he s ame formulas (3.4 1) - (3.43) as in the c ase of R ⋆ ( C × ) : ζ k ∗ ζ l = ( ζ l , k = l 0 , k 6 = l 1 O ⋆ ( C × ) = X n ∈ Z ζ n κ ( ζ k ) = X l ∈ Z ζ l ⊛ ζ k − l ε ( ζ k ) = ( 1 , k = 0 0 , k 6 = 0 σ ( ζ k ) = ζ − k Pr o of. F o r every seq uence of nonnega tiv e nu mbers r n > 0 satisfying the co ndition ∀ C > 0 X n ∈ Z r n · C | n | < ∞ (3.4 9) the set E r = { u ∈ O ( C × ) : ∀ n ∈ Z | u n | 6 r n } (3.50) is compact in O ( C × ) since it is clos ed and is co n tained in the rectang le f  , where f ( t ) = X n ∈ Z r n · | t | n , t ∈ C × Hence, E r generates a contin uous seminorm α 7→ max u ∈ E r |h u, α i| on O ⋆ ( C × ). But this is exactly the seminorm from (3.4 8 ): max u ∈ E r |h u, α i| = ma x | u | 6 r n      X n ∈ Z u n · α n      = X n ∈ Z r n · | α n | = ||| α ||| r It remains to verify that seminorms ||| · ||| r indeed gener ate the top ology of the space O ⋆ ( C × ). This follows from: Lemma 3.2. If p is a c ontinuous seminorm on O ⋆ ( C × ) , t hen the family of n umb ers r n = p ( ζ n ) satisfy c ondition (3.49) , and p is majorize d by t he seminorm ||| · ||| r : p ( α ) 6 ||| α ||| r (3.51) § 3. STEIN GROUPS AND HO PF ALGEBRAS CO NNECTED TO THEM 57 Pr o of. The se t D = { u ∈ O ( C × ) : sup α ∈O ⋆ ( C × ): p ( α ) 6 1 | α ( u ) | 6 1 } is compact in O ( C × ) generating the s eminorm p : p ( α ) = sup u ∈ D | α ( u ) | Therefore, r n = p ( ζ n ) = sup u ∈ D | ζ n ( u ) | = sup u ∈ D | u n | F o r a n y C > 0 we hav e: ∞ > sup u ∈ D || u || C +1 = sup u ∈ D X n ∈ Z | u n | · ( C + 1 ) | n | > sup u ∈ D sup n | u n | · ( C + 1 ) | n | = sup n  r n · ( C + 1 ) | n |  ⇓ ∃ M > 0 ∀ n ∈ Z r n 6 M ( C + 1) | n | ⇓ X n ∈ Z r n · C | n | 6 X n ∈ Z M ( C + 1) | n | · C | n | < ∞ i.e. r n indeed satisfy (3 .49). The form ula (3.51) is prov ed b y a sequence of inequalities a nalogous to (3.29). The c hain R ( C ) ⊂ O ( C ) ⊂ O ⋆ ( C ) ⊂ R ⋆ ( C ) . By sy m bo l R ( C ) we denote the usual algebra of p olyno- mials on the complex plane C . Let t k denote the monomial of degr ee k ∈ N o n C : t k ( x ) := x k , x ∈ C , k ∈ N (3.52) Every p olynomial u ∈ R ( C ) is uniquely represe n ted by the series (with ﬁnite num b er of nonzero terms) u = X k ∈ N u k · t k , u k ∈ C : card { k ∈ N : u k 6 = 0 } < ∞ (3.53) so the monomials t k form an algebr aic ba sis in the space R ( C ). The m ultiplication in R ( C ) is the usual m ultiplication o f p olynomia ls u · v = X k ∈ N n X i =0 u k − i · v i ! · t k and the top ology in R ( C ) is deﬁned a s the str ongest lo cally conv ex to polog y . This implies Prop osition 3.9. The mapping u ∈ R ( C ) 7→ { u k ; k ∈ N } ∈ C N is an isomorphism of top olo gic al ve ctor sp ac es R ( C ) ∼ = C N (3.54) 58 Remark 3. 1. W e can interpret formula (3.54) as isomorphis m of a lgebras, if the multiplication in C N is deﬁned by the same formula (1 .23) a s for a lgebra C G of p oint charges on the gro up G (deﬁned in § 1(c)), we s hould only remember here that the set N is not a g roup, but a monoid with r espect to the additive op eration used for this purp ose. On the other hand fo rm ula (3.54) is not a n isomor phism of Hopf algebr as, in particular , s ince C N is not a Hopf a lgebra at a ll with r espect to the op erations deﬁned in § 1(c): for x ∈ N the inv erse element x − 1 = − x do es not exist when x 6 = 0 , so the antipo de here ca nnot b e deﬁned by equality (1.27). Prop osition 3 .10. The algebr a R ( C ) of p olynomials on the c omplex plane C is a nucle ar Hopf-Br aun er algebr a with the top olo gy gener ate d by seminorms k u k r = X k ∈ N r k · | u k | , ( r k > 0) ( 3.55 ) and algebr aic op er ations deﬁne d on monomials t k by formulas t k · t l = t k + l 1 R ( C ) = t 0 (3.56) κ ( t k ) = k X i =0  k i  · t k − i ⊙ t i ε ( t k ) = ( 1 , k = 0 0 , k > 0 (3.57) σ ( t k ) = ( − 1) k · t k (3.58) Pr o of. The a lgebraic op erations which we did not ﬁnd yet – comultiplication, c ounit and a n tipo de – are computed by for m ulas (3 .5)-(3.7). F or instance, comultiplication: e κ ( t k )( x, y ) = t k ( x + y ) = ( x + y ) k = k X i =0  k i  · x k − i · y i = k X i =0  k i  · t k − i ⊡ t i ( x, y ) ⇓ e κ ( t k ) = k X i =0  k i  · t k − i ⊡ t i ⇓ κ ( t k ) = ρ C , C ( e κ ( t k )) = k X i =0  k i  · t k − i ⊙ t i F o llo wing the terminolo gy of [1], we ca ll a curr ent of de gr e e 0, or a curr ent on C a n arbitra ry linear functional α : R ( C ) → C on the s pace of po lynomials R ( C ) (every such a functional is automatically contin uous). F or m ula (3.54) implies that every co mpact set in the space o f p olynomials R ( C ) is ﬁnite- dimensional, hence it is contained in a convex hull of a ﬁnite set of ba sis monomia ls t k . Therefore the top ology of the space R ⋆ ( C ) of currents on C (whic h is forma lly deﬁned as the top ology of uniform conv ergence o n co mpact s ets in R ( C )) can be deﬁned a s the top ology of co n vergence on monomia ls t k , i.e. the topo logy generated by seminorms k α k N = X k ∈ N   α ( t k )   , where N is an arbitra ry ﬁnite set in N . The functional of taking deriv ative of the k -th deriv a tiv e in the p oint 0 is a typical example of a current: τ k ( u ) = d k d x k u ( x ) !      x =0 , u ∈ R ( C ) (3.5 9) § 3. STEIN GROUPS AND HO PF ALGEBRAS CO NNECTED TO THEM 59 By the T aylor theorem, these functionals are co nnected with the co eﬃcien ts u k in the deco mpositio n (3.53) of a p olynomial u ∈ R ( C ) thro ugh the for m ula u k = 1 k ! · τ k ( u ) (3.60) Therefore the ac tion of a cur ren t α ∈ R ⋆ ( C ) on a po lynomial u ∈ R ( C ) can b e written by for m ula α ( u ) = α X k ∈ N u k · t k ! = X k ∈ N u k · α ( t k ) = X k ∈ N 1 k ! · τ k ( u ) · α ( t k ) = X k ∈ N 1 k ! · α ( t k ) · τ k ! ( u ) (3.61) This means that α is decomp osed int o a serie s in terms o f τ k : α = X k ∈ N α k · τ k , α k = 1 k ! · α ( t k ) (3.6 2) W e c an deduce from this that curren ts τ k form a basis in t he top ological v ector spaces R ⋆ ( C ): every func- tional α ∈ R ⋆ ( C ) can be uniquely represented by a converging in R ⋆ ( C ) serie s (3.62), where co eﬃcients α k ∈ C contin uously dep end o n α ∈ R ⋆ ( C ). F r om (3.6 1) it follows that the action of the cur ren t α on a p olynomial u is deﬁned by for m ula α ( u ) = X k ∈ N 1 k ! · τ k ( u ) | {z } u k · 1 k ! · α ( t k ) | {z } α k · k ! = X k ∈ N u k · α k · k ! and the basis cur ren ts τ k act on monomials t k by formula τ k ( t n ) = ( 0 , n 6 = k n ! , n = k , This means that the system τ k is not a dual b asis for t k : it diﬀers fro m the dual basis by the co eﬃcients k !. How ev er, since R ⋆ ( C ) ∼ = C N (this follows from (3.54)) a nd, on the other hand, in C N any tw o bases are isomor phic (Theorem 0.20), we hav e Prop osition 3.11. The mapping α ∈ R ⋆ ( C ) 7→ { α k ; k ∈ N } ∈ C N is an isomorphism of top olo gic al ve ctor sp ac es: R ⋆ ( C ) ∼ = C N (3.63) This iso morphism is no t how ever, an is omorphism of algebr as, since the m ultiplication in R ⋆ ( C ) is not coo rdinate-wise (i.e. not by formula (1.2 0), as it could deﬁned in C N by ana logy with the case of § 1(c)), but as p ow er series: α ∗ β = X k ∈ N k X i =0 α k − i · β i ! · τ k , (3.64) This follows fro m formula (3.66) b elow: Prop osition 3. 12. The algebr a R ⋆ ( C ) of curr ents of zer o de gr e e on the c omplex plane C is a n ucle ar Hopf-F r ´ echet algebr a with t he t op olo gy gener ate d by seminorms k α k K = K X k =0 | α k | , ( K ∈ N ) (3.65) 60 and algebr aic op er ations deﬁne d on b asis elements τ k by formulas τ k ∗ τ l = τ k + l 1 R ⋆ ( C ) = τ 0 (3.66) κ ( τ k ) = k X i =0  k i  · τ k − i ⊛ τ i ε ( τ k ) = ( 1 , k = 0 0 , k > 0 (3.67) σ ( τ k ) = ( − 1) k · τ k (3.68) Pr o of. There are many wa ys to pr o ve these formulas, for insta nce, to prov e (3.66) o ne ca n use formula (3.57) for comultiplication in R ( C ): ( τ k ∗ τ l )( t m ) = ( τ k ⊛ τ l )( κ ( t m )) = (3.57) = ( τ k ⊛ τ l ) m X i =0  m i  · t m − i ⊙ t i ! = = m X i =0  m i  · τ k ( t m − i ) · τ l ( t i ) =     m i  · ( m − l )! · l ! , m = k + l 0 , m 6 = k + l    =  m ! , m = k + l 0 , m 6 = k + l  = τ k + l ( t m ) . (on each monomial t m the action of functionals τ k ∗ τ l and τ k + l coincide, hence they coincide themselves). Let us consider no w algebra O ( C ) of entire functions and its dial algebra O ⋆ ( C ) of analytical function- als on complex plane C . Again, let t k denote the monomia l of degree k ∈ N on C , and τ k the functional of taking k -th deriv ative in the p oin t 0: t k ( z ) = z k τ k ( u ) = d k d z k u ( z ) !      z =0  x ∈ C , u ∈ O ( C )  Then it is co n venien t to repr esen t elements O ( C ) and O ⋆ ( C ) as (conv erging in thes e s paces) s eries u ∈ O ( C ) ⇐ ⇒ u = ∞ X n =0 u n · t n , u n = 1 n ! τ n ( u ) ∈ C : ∀ C > 0 ∞ X n =0 | u n | · C n < ∞ (3.69) α ∈ O ⋆ ( C ) ⇐ ⇒ α = ∞ X n =0 α n · τ n , α n = 1 n ! α ( t n ) ∈ C : ∃ M , C > 0 ∀ n ∈ N | α n | 6 M · C n n ! (3.70) The action of an ana lytical functional α o n an e n tire function u is descr ibed by for m ula h u, α i = ∞ X n =0 u n · α n · n ! and the multiplications in O ( C ) and in O ⋆ ( C ) are deﬁned by the same fo rm ulas a s for the usual p ow er series: u · v = ∞ X n =0 n X i =0 u i · v n − i ! · t n , α ∗ β = X k ∈ N k X i =0 α i · β k − i ! · τ k , (3.71) Prop osition 3 . 13. The algebr a O ( C ) of entir e funct ions on c omplex plane C is a nu cle ar H opf-F r´ echet algebr a with the top olo gy gener ate d by seminorms k u k C = X k ∈ N | u k | · C k ( C > 1) (3.72) § 3. STEIN GROUPS AND HO PF ALGEBRAS CO NNECTED TO THEM 61 and algebr aic op er ations deﬁne d on monomials t k by t he same formulas (3.56) - (3.5 8 ) as for R ( C ) : µ ( t k ⊙ t l ) = t k + l 1 R ( C ) = t 0 κ ( t k ) = k X i =0  k i  · t k − i ⊙ t i ε ( t k ) = ( 1 , k = 0 0 , k > 0 σ ( t k ) = ( − 1) k · t k Pr o of. The formulas for a lgebraic op erations ar e proved similarly with the same formulas for R ( C ), so it r emains only to explain, why the top ology is generated b y s eminorms (3.72). This is a sub ject of mathematical folklo re (see e.g. [26]): clearly , every usual s eminorm | u | K = max z ∈ K | u ( z ) | , wher e K is a compact set in C , is ma jorize d by some seminorm || u || C , namely by the o ne with C = max z ∈ K | z | : | u | K = max z ∈ K | u ( z ) | = max z ∈ K      ∞ X n =0 u n · z n      6 ma x z ∈ K ∞ X n =0 | u n | · | z n | 6 ∞ X n =0 | u n | · C n = || u || C On the contrary , for every C > 0 we c an take a compact set K = { z ∈ C : | z | 6 C + 1 } , and then from formulas for Cauch y co eﬃcien ts | u n | 6 | u | K ( C + 1) n it follows that || u || C is sub ordinated to | u | K (with the constant C + 1 ): || u || C = ∞ X n =0 | u n | · C n 6 ∞ X n =0 | u | K ( C + 1) n · C n 6 | u | K · ∞ X n =0  C C + 1  n = | u | K · 1 1 − C C +1 = ( C + 1) · | u | K Prop osition 3. 14. The algebr a O ⋆ ( C ) of analytic al functionals on c omplex plane C is a nucle ar Hopf- Br aun er algebr a with the top olo gy gener ate d by seminorms ||| α ||| r = X k ∈ N r k · | α k | · k ! r k > 0 : ∀ C > 0 X k ∈ N r k · C k < ∞ ! (3.73) and algebr aic op er ations deﬁne d on b asis elements τ k by the same formulas (3.6 6) - (3.68) as in the c ase of R ⋆ ( C ) : µ ( τ k ⊛ τ l ) = τ k + l 1 R ⋆ ( C ) = τ 0 κ ( τ k ) = k X i =0  k i  · τ k − i ⊛ τ i ε ( τ k ) = ( 1 , k = 0 0 , k > 0 σ ( τ k ) = ( − 1) k · τ k Pr o of. Here a gain the formulas for alg ebraic op erations are pr o ved similarly with the case of R ⋆ ( C ), so we need only to expla in, why the top ology is g enerated by seminorms (3.73). Note ﬁrst that for any sequence of non-neg ativ e num b ers r k > 0, satisfying the co ndition ∀ C > 0 X k ∈ N r k · C k < ∞ (3.74 ) the set E r = { u ∈ O ( C ) : ∀ k ∈ N | u k | 6 r k } (3.75) 62 is compact in O ( C ), since it is clos ed and is co n tained in the rectang le f  , where f ( x ) = X k ∈ N r k · | x | k , x ∈ C Hence, E r generates a co n tin uous seminorm α 7→ max u ∈ E r |h u, α i| on O ⋆ ( C ). But this is exactly semino rm (3.73): max u ∈ E r |h u, α i| = ma x | u | 6 r k      X k ∈ N u k · α k · k !      = X k ∈ N r k · | α k | · k ! = ||| α ||| r It remains to verify that the seminor ms ||| · ||| r indeed genera te the top ology of the spa ce O ⋆ ( C × ). This follows from: Lemma 3.3. If p is a c ontinuous seminorm on O ⋆ ( C ) , then the numb ers r k = 1 k ! p ( τ k ) satisfy the c ondition (3.7 4 ) , and p is majorize d by the seminorm ||| · ||| r : p ( α ) 6 ||| α ||| r (3.76) Pr o of. The se t D = { u ∈ O ( C ) : sup α ∈O ⋆ ( C ): p ( α ) 6 1 | α ( u ) | 6 1 } is compact in O ( C ) and gener ates the seminor m p : p ( α ) = sup u ∈ D | α ( u ) | Hence r k = 1 k ! p ( τ k ) = 1 k ! sup u ∈ D | τ k ( u ) | = sup u ∈ D | u k | F o r ea c h C > 0 we hav e: ∞ > sup u ∈ D || u || C +1 = sup u ∈ D X k ∈ N | u k | · ( C + 1 ) k > sup u ∈ D sup k | u k | · ( C + 1 ) k = sup k  r k · ( C + 1 ) k  ⇓ ∃ M > 0 ∀ k ∈ N r k 6 M ( C + 1) k ⇓ X k ∈ N r k · C k 6 X k ∈ N M ( C + 1) k · C k < ∞ I.e., r k indeed satisfy co ndition (3.74). F or m ula (3 .76) is pr o ved by the chain o f inequalities ana logous to (3.29). Prop osition 3.15. The mappings t k 7→ t k 7→ τ k 7→ τ k ( k ∈ N ) deﬁne the chain of homomorphisms of rigid Hopf algebr as: R ( C ) → O ( C ) → O ⋆ ( C ) → R ⋆ ( C ) (3.77) § 4. FUNCTIONS OF EXPONENTIAL TYPE ON A STEIN GROUP 63 § 4 F unc tions of exp onen tial t yp e on a Stein group (a) Semic haracters and in v erse semic haracters on Stein groups Let G b e a Stein group. Then — a lo cally b ounded function f : G → [1 , + ∞ ) is called a semichar acter , if it satisﬁes the following so called submultiplic ativity ine quality : f ( x · y ) 6 f ( x ) · f ( y ) , x, y ∈ G (4.1) — a function ϕ : G → (0 ; 1], lo cally separated from zero , is ca lled inverse semichar acter , if it satisﬁes the inv erse inequa lit y: ϕ ( x ) · ϕ ( y ) 6 ϕ ( x · y ) , x, y ∈ G (4.2) Clearly , if f : G → [1 , + ∞ ) is a semicharacter, then the inv erse function ϕ ( x ) = 1 f ( x ) (4.3) is an inv erse s emic harac ter, and vice versa. Prop erties o f sem ic haracters and i nv e rse s emic haracters: (i) The set of all semicharacters o n G is closed under the following o pera tions: — multiplication by a s uﬃcien tly big constant: C · f ( C > 1), — multiplication: f · g , — addition: f + g , — taking maximum: max { f , g } . (ii) The set of all inv erse semicharacters on G is closed under the following op eratio ns: — multiplication by a s uﬃcien tly small consta n t: C · ϕ ( C 6 1), — multiplication: ϕ · ψ , — taking half of har monic mean: ϕ · ψ ϕ + ψ — taking minimum: min { ϕ, ψ } . Pr o of. Ha ving in mind the dua lit y b etw een semich ar acters and in verse s emic haracter s reﬂected by form ula (4.3), we can consider o nly the ca se of semicharacters. If f is a semicharacter on G a nd C > 1, then C · f ( x · y ) 6 C · f ( x ) · f ( y ) 6  C · f ( x )  ·  C · f ( y )  If f and g are semicharacters o n G , then considering their multiplication we have: ( f · g )( x · y ) = f ( x · y ) · g ( x · y ) 6 f ( x ) · f ( y ) · g ( x ) · g ( y ) = = f ( x ) · g ( x ) · f ( y ) · g ( y ) = ( f · g )( x ) · ( f · g )( y ) F o r their sum we hav e: ( f + g )( x · y ) = f ( x · y ) + g ( x · y ) 6 f ( x ) · f ( y ) + g ( x ) · g ( y ) 6 6 f ( x ) · f ( y ) + g ( x ) · f ( y ) + f ( x ) · g ( y ) + g ( x ) · g ( y ) =  f ( x ) + g ( x )  ·  f ( y ) + g ( y )  = ( f + g )( x ) · ( f + g )( y ) 64 The pro of for maximum max { f , g } is based on the following ev iden t ineq ualit y: max { a · b, c · d } 6 max { a, c } · max { b, d } ( a, b, c, d > 0) (4.4) It implies: max { f , g } ( x · y ) = max { f ( x · y ) , g ( x · y ) } 6 max { f ( x ) · f ( y ) , g ( x ) · g ( y ) } 6 (4 .4) 6 6 ma x { f ( x ) , g ( x ) } · max { f ( y ) , g ( y ) } = max { f , g } ( x ) · max { f , g } ( y ) Example 4.1 . All the s ubm ultiplicative matrix norms (see [1 5]), for insta nce || x || = n X i,j =1 | x i,j | , || x || = v u u t n X i,j =1 | x i,j | 2 (4.5) are semicharacter s on GL n ( C ). F ro m the pr oper ties of semicharacters it follows tha t the function o f the form r N C ( x ) = C · max {|| x || ; || x − 1 ||} N , C > 1 , N ∈ N (4.6) are aga in s emic haracter s on GL n ( C ). Prop osition 4.1. F or any submultiplic ative matrix n orm || · || on GL n ( C ) the semichar acters of t he form (4.6) majorize al l other s emichar acters on GL n ( C ) . Pr o of. Note from the very beginning that it is suﬃcient to consider the ca se when || · || is the Euclid nor m on the alg ebra M n ( C ) of all complex ma trices n × n : || x || = sup ξ ∈ C n : || ξ || 6 1 || x ( ξ ) || , where || ξ || = v u u t n X i =1 | ξ i | 2 , – s ince any o ther no rm o n M n ( C ) ma jorizes the Euclid seminor m up to a constant multiplier, this will prov e our pr opos ition. Consider the set K :=  x ∈ GL n ( C ) : max {|| x || ; || x − 1 ||} 6 2  =  x ∈ GL n ( C ) : ∀ ξ ∈ C n 1 2 · || ξ || 6 || x ( ξ ) || 6 2 · || ξ ||  It is clos ed and bo unded in the algebra of matric es M n ( C ), hence it is compa ct. This is a genera ting compact in GL n ( C ) GL n ( C ) = ∞ [ m =1 K m , K m = K · ... · K | {z } m fac tors , (4.7) In addition, K m =  x ∈ GL n ( C ) : max {|| x || ; || x − 1 ||} m 6 2 m  (4.8) Indeed, if x ∈ K m , then x = x 1 · ... · x m , where || x i || 6 2 and || x − 1 i || 6 2, hence || x || 6 m Y i =1 || x i || 6 2 m , || x − 1 || 6 m Y i =1 || x − 1 i || 6 2 m On the co n trary , suppos e max {|| x || m ; || x − 1 || m } 6 2 m . Consider the p olar decomp osition: x = r · u , where r is a p ositive deﬁnite Hermitian, and u a unitary matrices . Let us deco mpose r into a pro duct r = v · d · v − 1 , wher e v is unitary , and d a diago nal matric es: d =     d 1 0 ... 0 0 d 2 ... 0 ... ... ... ... 0 0 ... d n     , d i > 0 § 4. FUNCTIONS OF EXPONENTIAL TYPE ON A STEIN GROUP 65 The m -th ro ot m √ d =     m √ d 1 0 ... 0 0 m √ d 2 ... 0 ... ... ... ... 0 0 ... m √ d n     has the following pro perties : max i =1 ,...,n d i = || d || = || r || = || x || 6 2 m = ⇒ || m √ d || = max i =1 ,...,n m p d i 6 2 , max i =1 ,...,n d − 1 i = || d − 1 || = || r − 1 || = || x − 1 || 6 2 m = ⇒ || m √ d − 1 || = max i =1 ,...,n m q d − 1 i 6 2 As a coro llary , the matr ix y = v · m √ d · v − 1 belo ngs to K : max {|| y || ; || y − 1 ||} = max {|| m √ d || ; || m √ d − 1 ||} 6 2 = ⇒ y ∈ K Hence the matrix y · u also b elongs to K , and we o btain x = v · d · v − 1 · u = v · ( m √ d ) m − 1 · v − 1 · v · ( m √ d ) · v − 1 · u = y m − 1 | {z } ∈ K m − 1 · y · v |{z} ∈ K ∈ K m W e hav e prov ed formula (4.8). Now let f b e an arbitra ry s emic harac ter on GL n ( C ). Put C = sup x ∈ K f ( x ) , N > log 2 C and let us show that ∀ x ∈ GL n ( C ) f ( x ) 6 r N C ( x ) (4.9) T ake x ∈ GL n ( C ). F rom (4.7) it follows that x ∈ K m \ K m − 1 for so me m ∈ N . By formula (4.8) we hav e: 2 m − 1 < ma x {|| x || ; || x − 1 ||} 6 2 m ⇓ m − 1 < log 2  max {|| x || ; || x − 1 ||}  ⇓ f ( x ) 6 sup y ∈ K f ( y ) m 6 C m 6 C · C m − 1 < C · C log 2  max {|| x || ; || x − 1 ||}  = = C ·  max {|| x || ; || x − 1 ||}  log 2 C 6 C ·  max {|| x || ; || x − 1 ||}  N = r N C ( x ) In the sp ecial case when n = 1 we hav e: Corollary 4. 1. On the c omplex cir cle C × the semichar acters of the form r N C ( t ) = C · max {| t | ; | t | − 1 } N , C > 1 , N ∈ N majorize al l other s emicha r acters. 66 Prop osition 4. 2. If G is a c omp actly gener ate d S tein gr oup and K a c omp act neighb orho o d of identity in G , gener ating G , G = ∞ [ n =1 K n , K n = K · ... · K | {z } n factors , then for any C > 1 the rule h C ( x ) = C n ⇐ ⇒ x ∈ K n \ K n − 1 (4.10) deﬁnes a semichar acter h C on G . Such semichar acters form a fu n damental system among al l semichar- acters on G : every semichar acter f on G is majorize d by some semichar acter h C f ( x ) 6 h C ( x ) , x ∈ G, – for this the c onstant C should b e chosen such that C > ma x t ∈ K f ( t ) (4.11) Pr o of. The lo cal b oundedness of h C is obvious, so we need to chec k the submultiplicativit y inequality . T ake x, y ∈ G and cho ose k , l ∈ N such that x ∈ K k \ K k − 1 , y ∈ K l \ K l − 1 . Then x · y ∈ K k + l , so h C ( x · y ) 6 C k + l = C k · C l = h C ( x ) · h C ( y ) If now f is an arbitra ry s emic haracter , and C s atisﬁes the condition (4.11), then x ∈ K n \ K n − 1 = ⇒ f ( x ) 6  max t ∈ K f ( t )  n = C n = h C ( x ) (b) Subm ultiplicativ e rhom buses and dually subm ultiplicative rectangles Let us introduce the following deﬁnitio ns: — a clo sed a bsolutely conv ex neighborho o d o f zero ∆ in O ⋆ ( G ) is said to b e submu ltipli c ative , if for any functionals α, β fr om ∆ their convolution α ∗ β a lso b elong to ∆ : ∀ α, β ∈ ∆ α ∗ β ∈ ∆ shortly this is expres sed by the following inclusio n: ∆ ∗ ∆ ⊆ ∆ — an absolutely closed co mpact set D in O ( G ) is sa id to be dual ly submultiplic ative , if its p olar D ◦ is a submu ltiplicative neighborho o d of zer o: D ◦ ∗ D ◦ ⊆ D ◦ Lemma 4.1. (a) If ϕ : G → (0; 1 ] is an inverse semichar acter on G , then its rhombus ϕ  is a (close d, absolutely c onvex, and) submultiplic ative neighb orho o d of zer o in O ⋆ ( G ) . (b) If ∆ ⊆ O ⋆ ( G ) is a close d absolutely c onvex and submultiplic ative neighb orho o d of zer o in O ⋆ ( G ) , then its inner envelop e ∆ ♦ is an inverse semichar acter on G . § 4. FUNCTIONS OF EXPONENTIAL TYPE ON A STEIN GROUP 67 Pr o of. 1. Let us denote ε x = ϕ ( x ) · δ x ∈ O ⋆ ( G ) , then ϕ  = absconv { ϕ ( x ) · δ x ; x ∈ M } = a b sconv { ε x ; x ∈ M } (4.1 2) and the inclusion ϕ  ∗ ϕ  ⊆ ϕ  is veriﬁed in thr ee steps. First we should no te that ∀ x, y ∈ G ε x ∗ ε y ∈ ϕ  Indeed, ε x ∗ ε y = ϕ ( x ) · δ x ∗ ϕ ( y ) · δ y = ϕ ( x ) · ϕ ( y ) · δ x · y = ϕ ( x ) · ϕ ( y ) ϕ ( x · y ) | {z } 6 1 · ε x · y ∈ absco n v { ε z ; z ∈ M } = ϕ  Second, we take ﬁnite a bsolutely co n vex combinations of functionals ε x : α = k X i =1 λ i · ε x i , β = l X j =1 µ j · ε y j   k X i =1 | λ i | 6 1 , l X j =1 | µ j | 6 1   (4.13) F o r them we hav e: α ∗ β = k X i =1 λ i · ε x i ∗ l X j =1 µ j · ε y j = X 1 6 i 6 k, 1 6 j 6 l λ i · µ j | {z } ↑ P 1 6 i 6 k, 1 6 j 6 l | λ i · µ j | = P k i =1 | λ i | · P l j =1 | µ j | 6 1 · ε x i ∗ ε y j | {z } ∈ ϕ  ∈ ϕ  |{z} absolutely conv ex set And, third, for a rbitrary α, β ∈ ϕ  the inclusion α ∗ β ∈ ϕ  bec omes a co rollary of tw o facts: that func- tionals (4.13) are dense in the set ϕ  = absconv { ε x ; x ∈ M } , a nd that the co n volution ∗ is contin uous. 2. W e use her e formula (2.19): ∆ ∗ ∆ ⊆ ∆ = ⇒ ∀ x, y ∈ G ∆ ♦ ( x ) · δ x | {z } ∈ ∆, by (2.19) ∗ ∆ ♦ ( y ) · δ y | {z } ∈ ∆, by (2.19) = ∆ ♦ ( x ) · ∆ ♦ ( y ) · δ x · y ∈ ∆ = ⇒ = ⇒ ∆ ♦ ( x ) · ∆ ♦ ( y ) 6 max { λ > 0 : λ · δ x · y ∈ ∆ } = (2.19) = ∆ ♦ ( x · y ) = ⇒ ∆ ♦ ( x ) · ∆ ♦ ( y ) 6 ∆ ♦ ( x · y ) Lemma 4.2. (a) If f : G → [1; ∞ ) is a semichar acter on G , t hen its r e ct angle f  is dual ly submu ltiplic ative. (b) If D ⊆ O ( G ) is a dual ly submu ltiplic ative absolutely c onvex c omp acts set , t hen its outer envelop e D  is a semichar acter on G . 68 Pr o of. 1. If f : G → [1; ∞ )] is a semicharacter, then 1 f is a n inverse semicharacter, therefor e by Lemma 4.1 (a) the rho m bus  1 f   = (2.28) = ( f  ) ◦ is a submultiplicativ e neighbor hoo d o f zer o. This means that f  is dually subm ultiplicative. 2. If D ⊆ O ( G ) is a dually subm ultiplicative absolutely con vex compact set, then its p olar D ◦ ⊆ O ⋆ ( G ) is a submultiplicativ e neighbor hoo d o f zer o in O ⋆ ( G ), hence by Lemma 4.1 (b) the inner env elop e ( D ◦ ) ♦ = (2.29) = 1 D  is an inv erse s emic harac ter. Therefore, D  m ust b e a semicharacter. Lemmas 4.1 and 4.2 together with formulas ∆ = ∆ ♦ and D = D  for r hom buses and r ectangles give the following Theorem 4.1. (a) A rhombus ∆ in O ⋆ ( G ) is submultiplic ative if and only if its inner envelop e ∆ ♦ is an inverse semichar acter on G . (b) A r e ctangle D in O ( G ) is dual ly su bmultiplic ative if and only if its out er envelop e D  is a semichar- acter on G . The following result shows that the submultiplicativ e rhombuses form a fundament al s ystem among all submultiplicativ e closed absolutely convex neighbo rho od of zero in O ⋆ ( G ): Theorem 4.2. (a) Every close d absolutely c onvex neighb orho o d of zer o ∆ in O ⋆ ( G ) c ontains some submultiplic ative rhombus, namely ∆ ♦  . (b) Every dual ly submultiplic ative absolutely c onvex c omp act set D in O ( G ) is c ontaine d in some dual ly submultiplic ative r e ctangle, namely in D  . Pr o of. 1. If ∆ is a closed a bsolutely c on vex neighbor hoo d of zero in O ⋆ ( G ), then by Lemma 4.1(b), ∆ ♦ is an inv erse s emic harac ter, hence by Lemma 4.1(a), the r hom bus ∆ ♦  is subm ultiplicative. 2. If D is a dually submult iplicative abso lutely conv ex compa ct set in O ( G ), then its p olar D ◦ is a clo sed absolutely conv ex subm ultiplicative neighborho o d of zero in O ⋆ ( G ), and by what we have already prov ed ( D ◦ ) ♦  is a submultiplicative rhombus. I.e. the set ( D  ) ◦ = ( D ◦ ) ♦  is submult iplicative. Therefore the re ctangle D  is dually submultiplicativ e. In a ccordance with the deﬁnitions of § 2, we call a function f on G an enveloping semichar acter , if it is an outer env elop e a nd at the same time a semicharacter on G . Theorem 4 .3. If G is a c omp actly gener ate d Stein gr oup, then the systems of al l semichar acters, all enveloping semichar acters and al l dual submultiplic ative r e ctangles in G c ontain c ountable c oﬁnal su bsys- tems: – ther e exists a se quenc e h N of semichar acters on G such t hat every semichar acter g is majorize d by some semichar acter h N : g ( x ) 6 h N ( x ) , x ∈ G – ther e exists a se quenc e f N of enveloping semichar acters on G such that every enveloping s emichar- acter g is majorize d by some semichar acter f N : g ( x ) 6 f N ( x ) , x ∈ G § 4. FUNCTIONS OF EXPONENTIAL TYPE ON A STEIN GROUP 69 – ther e exists a se quenc e E N of dual ly submupltiplic ative r e ctangles in G su ch that every dual ly sub- multiplic ative r e ctangle D in G is c ontaine d in some E N : D ⊆ E N Pr o of. This follows from Pro position 4.2: the semic hara cters h N , N ∈ N , deﬁned in (4.10) , are the sequence we lo ok for. The sequences E N and f N are deﬁned as follows: E N = { u ∈ O ( G ) : max x ∈ K n | u ( x ) | 6 N n } = ( h N )  (4.14) f N = ( E N )  = ( h N )  (4.15) ( E N are dually submultiplicativ e rectang les by Lemma 4.2(a), and f N are env eloping semicharacters b y Lemma 4.2(b)). If now D is a dually submultiplicativ e rectangle, then by Lemma 4 .2(b), its o uter env elop e D  is a semicharacter, hence by Prop osition 4 .2, there is N ∈ N such that D  6 h N This means that D is contained in some E N : D = ( D  )  ⊆ ( h N )  = E N This in its turn implies tha t D  6 ( E N )  = f N , – so e v ery env eloping semicharacter (always being of the form D  by Theor em 4.1) is ma jorized by some f N . (c) Holomorphic functions of exp onen t ial t yp e Algebra O exp ( G ) of holomorphic functions of expo nen tial t ype . A holo morphic function u ∈ O ( G ) o n a compactly g enerated Stein group G we call a function of exp onential typ e , if it is b ounded by some semicharacter: | u ( x ) | 6 f ( x ) , x ∈ G  f ( x · y ) 6 f ( x ) · f ( y )  The set of all ho lomorphic functions of exp onential t yp e on G is denoted b y O exp ( G ). It is a subspace in O ( G ) and, by The orem 4.2, O exp ( G ) can b e considered a s the union of all dually submultip licative rectangles in O ( G ): O exp ( G ) = [ D is a dually submultiplicative rectangle in O ( G ) D = [ f is a semicharacter in G f  or, what is the same, the union o f all s ubspaces o f the for m C D , where D is a dually s ubm ultiplicative rectangle in O ( G ): O exp ( G ) = [ D is a dually submultiplicative rectangle in O ( G ) C D (4.16) This equality allows to endow O exp ( G ) with the na tural top olog y – the to polog y of injective (lo cally conv ex) limit of Smith spaces C D : O exp ( G ) = lim → D is a dually submultiplicative rectangle in O ( G ) C D (4.17) 70 F r om Theorem 4 .3 it follows that in this limit the sys tem of a ll dually submultiplicativ e r ectangles can be repalced by some co un table subsystem: O exp ( G ) = lim − → N →∞ C E N (4.18) T ogether with Theorem 0.6 this gives the following fact. Theorem 4.4 . The sp ac e O exp ( G ) of t he functions of exp onent ial gr owth on a c omp actly gener ate d Stein gr ou p G is a Br auner sp ac e. Corollary 4.2. If G is a c omp actly gener ate d Stein gr oup, then every b ounde d set D in O exp ( G ) is c ont aine d in some re ctangle of t he form f  , wher e f is some semichar acter on G : D ⊆ f  Pr o of. By Prop osition 0.1, D is co n tained in one o f the compact sets E N . This set, being a dually submu ltiplicative r ectangle, by Theorem 4 .1 has the form f  N for some s emic haracter f N , namely for f N = E  N . Theorem 4.5 . The sp ac e O exp ( G ) of t he functions of exp onent ial gr owth on a c omp actly gener ate d Stein gr ou p G is a pr oje ctive st er e otyp e algebr a with r esp e ct t o u sual p ointwise multiplic ation of functions. Pr o of. Note that if tw o functions u, v ∈ O ( G ) are b ounded by semicharacters f and g , then their multi- plication u · v is b ounded by the se mic haracter f · g : u ∈ f  , v ∈ g  = ⇒ u · v ∈ ( f · g )  In o ther words, the multiplication ( u, v ) 7→ u · v in the s pace O ( G ) turns any compact of the form f  × g  (where f and g a re semicharacters) into the rectangle ( f · g )  . ( u, v ) ∈ f  × g  7→ u · v ∈ ( f · g )  Since this o pera tion is co n tinuous in O ( G ), it turns f  × g  int o ( f · g )  contin uously . On the other hand, ( f · g )  , b eing a dually submult iplicative rectang le, is contin uously included into the s pace O exp ( G ). Thus we obtain a contin uous mapping ( u, v ) ∈ f  × g  7→ u · v ∈ O exp ( G ) If now D and E are arbitrar y compac t sets in the Bra uner space O exp ( G ), then by Cor ollary 4.2, they are contained in so me compa ct sets of the for m f  and g  , where f and g ar e semicharacters. Hence, D × E ⊆ f  × f  F r om this w e ca n conclude that the o pera tion of multiplication is co n tinuous from D × E into O exp ( G ): ( u, v ) ∈ D × E ⊆ f  × g  7→ u · v ∈ O exp ( G ) This is true fo r a rbitrary compact s ets D and E in O exp ( G ). So we ca n a pply [1, Theorem 5 .24]: since O exp ( G ), b eing a Brauner space, is co-complete (i.e. its stereotype dual space is complete), b y [1, Theorem 2.4] it is saturated. This implies that any bilinea r fo rm o n O exp ( G ), if contin uous on compact sets of the form D × E , will b e co n tinuous O exp ( G ) in the sense of deﬁnition [1, § 5 .6]. Being applied to the op eration ( u, v ) 7→ u · v this means that the multiplication is contin uous in the sens e of conditions of Prop osition 1.1. Hence, O exp ( G ) is a pro jective stereotype algebr a. Let us note the following obvious fact: Theorem 4.6. If H is a close d sub gr oup in a Stein gr oup G , then the re striction of any holomorphic function of exp onential typ e on G t o H is a holomorphic function of exp onential typ e: u ∈ O exp ( G ) = ⇒ u | H ∈ O exp ( H ) (4.19) § 4. FUNCTIONS OF EXPONENTIAL TYPE ON A STEIN GROUP 71 Algebra O ⋆ exp ( G ) of exp onen tial analytic functionals . W e shall call the elemen ts of t he dual stereo- t yp e s pace O ⋆ exp ( G ), i.e. linear contin uous functionals on O exp ( G ) exp onent ial analytic functionals on the group G . The space O ⋆ exp ( G ) is called the sp ac e of exp onential analytic funct ionals on G . F r om Theo rem 4.4 we have Theorem 4 .7. The sp ac e O exp ( G ) of exp onential funct ionals on a c omp actly gener ate d Stein gr oup G is a F r´ echet sp ac e. Theorem 4 .8. The sp ac e O ⋆ exp ( G ) of exp onential functionals on a c omp actly gener ate d Stein gr ou p G is a pr oje ctive ster e otyp e algebr a with r esp e ct to usual c onvolution of functionals deﬁne d by F ormulas (3.10) - (3.1 4 ) : ( α, β ) 7→ α ∗ β Pr o of. 1. T ake a function u ∈ O exp ( G ) and note that for any p oint s ∈ G its s hift u · s (deﬁned by (3.10) ) is aga in a function from O exp ( G ): ∀ s ∈ G ∀ u ∈ O exp ( G ) u · s ∈ O exp ( G ) Indeed, if we take a semicharacter f ma jor izing u , we obta in: ∀ t ∈ G | ( u · s )( t ) | = | u ( s · t ) | 6 f ( s · t ) 6 f ( s ) · f ( t ) i.e., u ∈ f  = ⇒ u · s ∈ f ( s ) · f  (4.20) 2. Let us denote the mapping s 7→ u · s by b u , b u : G → O exp ( G )    b u ( s ) := u · s, s ∈ G and show that it is c on tinuous. Let s i be a sequence of p oints in G , tending to a p oint s : s i G − → i →∞ s Then the sequence of holomorphic functions b u ( s i ) ∈ O ( G ) tends t o the holomorphic function b u ( s ) ∈ O ( G ) uniformly on every compa ct se t K ⊆ G , i.e. in the space O ( G ): b u ( s i ) O ( G ) − → i →∞ b u ( s ) A t the s ame time, fr om (4.20) it follows that all those functions are b ounded by the semicharacter C · f , where C = max { sup i f ( s i ) , f ( s ) } (this v a lue is ﬁnite since the sequence s i with its limit s forms a co mpact set), so it lies in the rectang le g enerated by the semicharacter C · f : b u ( s i ) = u · s i ∈ C · f  , b u ( s ) = u · s ∈ C · f  In other words, b u ( s i ) tends to b u ( s ) in the co mpact set C · f  b u ( s i ) = u · s i C · f  − → i →∞ u · s = b u ( s ) Hence, b u ( s i ) tends to b u ( s ) in the space O exp ( G ): b u ( s i ) O exp ( G ) − → i →∞ b u ( s ) 3. The contin uity of the mapping b u : G → O exp ( G ) implies that fo r any functional β ∈ O ⋆ exp ( G ) the function β ◦ b u : G → C is holo morphic. W e can us e the Mor era theorem to prove this: cons ider a 72 closed or ien ted hyper surface Γ in G with a suﬃciently small dia meter so that int egr al ov er Γ of every holomorphic function v anishes, and show that the integral of β ◦ b u also v a nishes: Z Γ ( β ◦ b u )( s ) d s = 0 (4.21) Indeed, ta k e a net of functionals { β i ; i → ∞} ⊂ O ⋆ exp ( G ) approximating β in O ⋆ exp ( G ), and having a form of linear combinations of de lta-functionals: β i = X k λ k i · δ a k i , β i O ⋆ exp ( G ) − → i →∞ β Then we obtain the fo llo wing: since b u : G → O exp ( G ) is contin uous, β ◦ b u C ( G ) ← − ∞← i β i ◦ b u F r om this it follows that for any Ra don meas ure α ∈ C ( G ) α ( β ◦ b u ) ← − ∞← i α ( β i ◦ b u ) In particular for the functional of integration ov er the hypersur face Γ we hav e Z Γ ( β ◦ b u )( s ) d s ← − ∞← i Z Γ ( β i ◦ b u )( s ) d s = Z Γ X k λ k i · δ a k i ◦ b u ! ( s ) d s = = X k λ k i · Z Γ  δ a k i ◦ b u  ( s ) d s = X k λ k i · Z Γ δ a k i ( b u ( s )) d s = = X k λ k i · Z Γ b u ( s )( a k i ) d s = X k λ k i · Z Γ u ( s · a k i ) d s | {z } k 0 , since u is holom orp hic = 0 So indeed (4.2 1 ) is tr ue. 4. W e hav e showed that for any functional β ∈ O ⋆ exp ( G ) the function β ◦ b u : G → C is holomorphic. Let us show now that this function is of exp onential type: ∀ u ∈ O exp ( G ) ∀ β ∈ O ⋆ exp ( G ) β ◦ b u ∈ O exp ( G ) (4.22) Indeed, since the functional β ∈ O ⋆ exp ( G ) is b ounded on the compact set f  ⊆ O exp ( G ), it is a bo unded functional on the Bana c h representation of the Smith spa ce C f  , i.e. the following inequality ho lds: ∀ v ∈ C f  | β ( v ) | 6 M · k v k f  (4.23) where M = k β k ( f  ) ◦ := max v ∈ f  | β ( v ) | , k v k f  := inf { λ > 0 : v ∈ λ · f  } Now from formula (4.20) we have ∀ s ∈ G u · s ∈ f ( s ) · f  = ⇒ k u · s k f  := inf { λ > 0 : u · s ∈ λ · f  } 6 f ( s ) = ⇒ = ⇒ | ( β ◦ b u )( s ) | = | β ( u · s ) | 6 M · k u · s k f  6 M · f ( s ) So the function β ◦ b u is b ounded by the semicharacter M · f = k β k ( f  ) ◦ · f : u ∈ f  = ⇒ β ◦ b u ∈ k β k ( f  ) ◦ · f  (4.24) § 4. FUNCTIONS OF EXPONENTIAL TYPE ON A STEIN GROUP 73 5. Now let us note that ( β ◦ b u )( s ) = β ( u · s ) = (3.13) = ( u ∗ e β )( s ) , s ∈ G In o ther words, we prov ed that the function u ∗ e β = β ◦ b u b elongs to the spa ce O exp ( G ), so for any functional α ∈ O ⋆ exp ( G ) the convolution α ∗ β ( u ) = (3.14) = α ( u ∗ ˜ β ) is deﬁned. It remains to verify that the operatio n ( α, β ) 7→ α ∗ β is a con tinuous bilinear form, i.e. satisﬁes the conditions (i) a nd (ii) o f Pro positio n 1 .1. Let α i be a net tending to zero in O exp ( G ), and B a compact s et in O ⋆ exp ( G ). Let us consider an arbitrar y compact set D in O exp ( G ). By Cor ollary 4.2, it must b e contained in a rec tangle f  , where f is a semicharacter: D ⊆ f  On the other hand, on the Smith space C f  the norms of functionals β ∈ B ar e b ounded: sup β ∈ B k v k f  = M < ∞ So by virtue of (4.24) we obtain tha t ∀ u ∈ D ∀ β ∈ B u ∗ e β = β ◦ b u ∈ k β k ( f  ) ◦ · f  ⊆ M · f  Thu s the set { u ∗ e β ; u ∈ D , β ∈ B } is co n tained in the compact s et M · f  in the space O ⋆ exp ( G ). Hence, the net of functionals α i tends to zero uniformly on this set: ( α i ∗ β )( u ) = α i ( u ∗ e β ) C ⇒ i →∞ 0  u ∈ D , β ∈ B  This is true for an y compact set D , so the net α i ∗ β tends to zero in the spa ce O ⋆ exp ( G ) uniformly by β ∈ B : α i ∗ β O ⋆ exp ( G ) ⇒ i →∞ 0  β ∈ B  W e can omit the cas e of the inv erse seq uence of multipliers due to the identit y ] α ∗ β = e β ∗ e α (d) Examples. Finite g roups. As we hav e noted in § 3(a ), every ﬁnite group G , considered as a zero-dimensio nal complex manifold, is a linear complex Lie gr oup on which every function is holomorphic. On the other hand, every function u : G → C is b ounded (since its set of v a lues is ﬁnite), so u must b e a function of exp onen tial type. Thus, algebra s O exp ( G ) and O ( G ) in this ca se co incide with e ac h other and ar e equal to the alg ebra C G of all functions on G (with the p oint wise algebraic op erations and the top ology of po in twise co n vergence): O exp ( G ) = O ( G ) = C G 74 Groups C n . F or the case of complex plane C our deﬁnition, certainly , coincide with the classic al one – functions of exp onent ial type on C ar e en tire functions u ∈ O ( C ) growing not faster than the exp onen tial: ∃ A > 0 : | u ( x ) | 6 A | x | ( x ∈ C ) . According to the class ical theorems on the gr o wth of entire functions [22, 33], this is equiv alent to the condition that the der iv atives of u in a ﬁxed p oin t, say in zer o, grow not faster than the ex ponential: ∃ B > 0 : | u ( k ) (0) | 6 B k ( k ∈ N ) . The same is true for s ev eral v ariables: functions of exp onential t ype on C n according to our deﬁnition will b e exactly the functions u ∈ O ( C n ) satisfying the condition ∃ A > 0 : | u ( x ) | 6 A | x | ( x ∈ C n ) , (4.25) which turns out to b e equiv alent to the condition ∃ B > 0 : | u ( k ) (0) | 6 B | k | ( k ∈ N n ) , (4.26) where the factorial k ! and the mo dulus | k | of a multiindex k = ( k 1 , ..., k n ) ∈ N n are deﬁned b y the equalities k ! := k 1 · ... · k n , | k | := k 1 + ... + k n The equiv a lence of (4.25) to our deﬁnition follows fro m P rop osition 4.2, and the equiv alence betw een (4.25) and (4.26) is prov ed similarly with the case of one v ar iable: the implication (4.26) ⇒ (4 .25) is obvious, and the inv erse implication (4.25) ⇒ (4.2 6) is a cor ollary of the Cauch y inequalities (see [34]) for the co eﬃcients c k = u ( k ) (0) k ! of the T aylor series for the function u : ∀ R > 0 | c k | 6 max | x | 6 R | u ( x ) | R | k | 6 (4 .25) 6 A R R | k | ⇓ | c k | 6 min R> 0 A R R | k | = A R R | k |      R = | k | log A = A | k | log A | k | | k | · (log A ) | k | = 1 | k | | k | · B | k | , B = A 1 log A · log A ⇓ | u ( k ) (0) | = k ! · | c k | 6 k ! | k | | k | · B | k | 6 B | k | Groups GL n ( C ) . On the group GL n ( C ) the functions of exp onential type are exa ctly po lynomials, i.e. the functions of the fo rm u ( x ) = P ( x ) (det x ) m , x ∈ GL n ( C ) , m ∈ Z + (4.27) where P is a usual po lynomial o n matr ix elements of the matrix x , and det x is its determinant. Thus, the equality holds: O exp ( GL n ( C )) = R ( GL n ( C )) (4.28) Pr o of. 1. First, let us prove the inclusion R ( GL n ( C )) ⊆ O exp ( GL n ( C )). Note for this that every matrix element x 7→ x k,l § 4. FUNCTIONS OF EXPONENTIAL TYPE ON A STEIN GROUP 75 is a function of exponential t yp e, since it is b ounded by , fo r instance the ﬁrst of the matrix norms in (4.5): | x k,l | 6 n X i,j =1 | x i,j | = || x || As a corollar y , ev ery polyno mial x 7→ P ( x ) of matrix elements is also a function of expo nen tial t ype (since functions of exp onential type for m an algebr a). F ur ther, every p o wer of the determina n t, x 7→ (det x ) − m , is mult iplicative (det( x · y )) − m = (det x ) − m · (det y ) − m so its absolute v alue is also multip licative:   (det( x · y )) − m   =   (det x ) − m   ·   (det y ) − m   Therefore, the function x 7→ | (det x ) − m | is a semicharacter on GL n ( C ), and x 7→ (det x ) − m a function of exp onen tial type on GL n ( C ). Now, if w e multiply tw o functions o f exp onen tial t yp e x 7→ P ( x ) and x 7→ (det x ) − m , we o btain a function of exp onential type x 7→ P ( x ) / (det x ) m . 2. Let us prove the inverse inclusion: O exp ( GL n ( C )) ⊆ R ( GL n ( C )). If u is a holomo rphic function o f exp onen tial type on GL n ( C ), then by Prop osition 4.1, it must b e b ounded by some semicharacter of the form (4.6). In particular , fo r so me C > 1 a nd N ∈ N | u ( x ) | 6 C · max {|| x || ; || x − 1 ||} N , || x || = n X i,j =1 | x i,j | (4.29) The elements of the inv erse matrix x − 1 are obtained from x as complementary minor s, div ided by the determinant, s o we can treat them as p olynomials (o f degr ee n − 1) o f x i,j and (det x ) − 1 . This means that one ca n estimate the r igh thand side of (4.29) by a p olynomial (of degr ee N ( n − 1)) of | x i,j | and | det x | − 1 with nonnegative co eﬃcients: | u ( x ) | 6 C · ma x {|| x || ; || x − 1 ||} N 6 C · P  {| x i,j |} 1 6 i,j 6 n , | det x | − 1  Hence if we m ultiply u by (det x ) N ( n − 1) , we obtain a holomor phic function on GL n ( C ) bo unded b y a po lynomial of | x i,j | :    u ( x ) · (det x ) N ( n − 1)    6 C · Q  {| x i,j |} 1 6 i,j 6 n  Such a function is lo cally b ounded in the p oint s of the analytical set det x = 0, s o by the Riemann extension theorem (see [34]), it can be extended to a holo morphic function to the manifold M n ( C ) (of all complex matrices ). There fore, we can consider u ( x ) · (det x ) N ( n − 1) as a holomorphic function on M n ( C ) = C n 2 . Since it has a p olynomial growth, it must b e a p olynomial q of matrix elements x i,j : u ( x ) · (det x ) N ( n − 1) = q ( x ) Hence u ( x ) = q ( x ) (det x ) N ( n − 1) , i.e. u ∈ R ( GL n ( C )). Corollary 4. 3 . On the c omplex cir cle C × the functions of exp onential typ e ar e exactly the L aur ent p olynomials: u ( t ) = X | n | 6 N u n · t n , N ∈ N 76 (e) Injection ♭ G : O exp ( G ) → O ( G ) W e need to denote the injection of O exp ( G ) into O ( G ) by some sy m bo l. Let us use fo r this the symbo l ♭ G : ♭ G : O exp ( G ) → O ( G ) (4.30) This mapping is alwa ys injective, a ho momorphism of algebra s, and by deﬁnition of top ology in O exp ( G ), alwa ys c on tinuous. Below in Theo rem 4 .9 we s hall se e that if G is a linear group, then ♭ G has dense image in O ( G ). F r om equalit y (4.28) a nd Theo rem 4.6 it follows that if G is an arbitrary linear group (with a g iv en rep- resentation as a closed subgro up in GL n ( C )), then every function G that can b e extended to a p olynomial on GL n ( C ), is a function of ex ponential type. Thus, we hav e the following c hain of inclusions: R ( G ) ⊆ O exp ( G ) ⊆ O ( G ) (4.31) (here R ( G ) denotes the functions that can b e extended to p olynomials o n GL n ( C ) – we need this s peci- ﬁcation since G is not nec essarily a n algebr aic gro up). Theorem 4.9. If G is a line ar c omplex gr oup, t hen the algebr a O exp ( G ) of holomorphic functions of exp onential typ e on G is dense in t he algebr a O ( G ) of al l holomorphic fun ctions on G . Pr o of. Let ϕ : G → GL n ( C ) b e a holo morphic embedding a s a closed s ubgroup. By one o f the cor ollaries from the Car tan theor em [38 , 11.5.2], every holomorphic function v ∈ O ( G ) ca n b e extended to a holomorphic function u ∈ O ( GL n ( C )). Let us approximate u uniformly on compact set by po lynomials u i ∈ R ( GL n ( C )). By (4.28), all p olynomials u i are functions o f e xponential type o n GL n ( C ), hence their r estrictions u i | G are functions of exp onential type on G . Thus, v is a pproximated by functions of exp onen tial type u i | G uniformly on compact sets in G . (f ) N uclearit y of the spaces O exp ( G ) and O ⋆ exp ( G ) Theorem 4.10. F or any c omp actly gener ate d Stein gro up G the sp ac e O exp ( G ) is a nu cle ar Br auner sp ac e, and its dual sp ac e O ⋆ exp ( G ) a nucle ar F r´ echet s p ac e. W e premise the pro of of this fact b y tw o le mmas. The ﬁrst of them is tr ue for ar bitrary complex manifold and is prov ed by the same technique that is a pplied for the pro of of nuclearity of O ( C ) (see [25, 6.4.2]): Lemma 4.3. If M is a c omplex manifold, and K and L ar e two c omp act sets in M , such that K is strictly c ontaine d in L , K ⊆ Int L then ther e exists a c onstant C > 0 and a pr ob abili ty me asur e µ on L su ch that for any u ∈ O ( G ) we have | u | K 6 C · Z L | α ( u ) | µ ( d α ) (4.32) As a c or ol lary, the op er ator u | L 7→ u | K of r estriction is absolutely summing, and its quasinorm of absolute summing is not gr e ater than C : for any u 1 , ..., u l ∈ O ( M ) l X i =1 | u i | K 6 C · Z L l X i =1 | α ( u i ) | µ ( d α ) 6 C · sup α ∈ absconv ( δ L ) l X i =1 | α ( u i ) | Remark 4 .1. Here absconv  δ L  means the universal compact se t in the Smith spa ce C ⋆ ( L ) dual to the Banach spa ce C ( L ) o f contin uous functions on L , o r what is the s ame, the unit ba ll in the Banach space M ( L ) of Radon measures on L . W e use this notation, beca use it is conv enient to denote b y § 4. FUNCTIONS OF EXPONENTIAL TYPE ON A STEIN GROUP 77 δ L = { δ x ; x ∈ L } the set o f a ll delta-functionals o n C ( L ) – then the p olar B ◦ of the unit ball B in C ( L ) coincides with the abso lutely conv ex hu ll of δ L : B ◦ = absconv  δ L  = { α ∈ M ( L ) : || α || 6 1 } (the closur e with resp ect to top ology of C ⋆ ( L )). Lemma 4.4. F or any gener ating c omp act neighb orho o d of identity K in G G = ∞ [ n =1 K n ther e ar e c onst ants C > 0 , λ > 0 s u ch that for any l ∈ N and for arbitr ary u 1 , ..., u l ∈ O ( G ) , n ∈ N the fol lowing ine qu ality holds: l X i =1 | u i | K n 6 C · λ n − 1 · sup α ∈ absconv ( δ K 2 n +1 ) l X i =1 | α ( u i ) | (4.33) Pr o of. 1. The set U = Int K is an op en neig h b orho od o f identit y in G , s o the sys tem o f shifts { x · U ; x ∈ K 2 } is an op en c o vering of the co mpact set K 2 . Let us cho ose a ﬁnite subc o vering, i.e. a ﬁnite set F ⊆ K 2 such that K 2 ⊆ [ x ∈ F x · U = F · U Then we obtain: K n ⊆ F n − 1 · K ⊆ F n − 1 · K 2 ⊆ K 2 n +1 , n ∈ N (4.34) Here is the pro of: ﬁrst we should note that F ⊆ K 2 = ⇒ F n ⊆ K 2 n , n ∈ N After that in the seq uence (4.34) it is suﬃcient to chec k only the ﬁrst inclusion: K n ⊆ F n − 1 · K , n ∈ N (4.35) This is done by the induction: for n = 2 we hav e K 2 ⊆ F · U ⊆ F · K and, if (4.35) is true for some n , then for n + 1 we hav e: K n +1 = K n · K ⊆ F n − 1 · K · K ⊆ F n − 1 · K 2 ⊆ F n − 1 · F · K = F n · K 2. Since the compact set K 2 strictly contains K , by Lemma 4.3 there are C , µ such that | u | K 6 C · Z K 2 | α ( u ) | µ ( d α ) (4.36) Under the shift by an element x ∈ G this inequality takes form | u | x · K 6 C · Z x · K 2 | α ( u ) | ( x · µ )( d α ) (4.37 ) where x · µ is a shift of the meas ure µ : ( x · µ )( u ) = µ ( u · x ) , ( u · x )( t ) = u ( x · t ) 78 This implies that if E is an arbitra ry ﬁnite set in G , then the sum ν = 1 card E X x ∈ E x · µ is a proba bilit y mea sure on the set E · K 2 , with the prop erty | u | E · K 6 X x ∈ E | u | x · K 6 C · X x ∈ E Z x · K 2 | α ( u ) | ( x · µ )( d α ) = = C · X x ∈ E Z E · K 2 | α ( u ) | ( x · µ )( d α ) = C · Z E · K 2 | α ( u ) | X x ∈ E x · µ ! ( d α ) = = C · card ( E ) · Z E · K 2 | α ( u ) | ν ( d α ) F r om this we deduce that the restriction ope rator u | E · K 2 7→ u | E · K is absolutely summing, and the quasinorm of absolute summing can be estimated by the co nstan t C · card ( E ): for a ll u 1 , ..., u l ∈ O ( G ) l X i =1 | u i | E · K 6 C · card ( E ) · Z E · K 2 l X i =1 | α ( u i ) | ν ( d α ) 6 C · card ( E ) · sup α ∈ absconv ( δ E · K 2 ) l X i =1 | α ( u i ) | Now from formulas (4.3 4) we hav e: l X i =1 | u i | K n 6 l X i =1 | u i | F n − 1 · K 6 C · ca rd ( F n − 1 ) · sup α ∈ absconv ( δ F n − 1 · K 2 ) l X i =1 | α ( u i ) | 6 6 C · card ( F n − 1 ) · sup α ∈ absconv ( δ K 2 n +1 ) l X i =1 | α ( u i ) | 6 C · ( card ( F )) n − 1 · sup α ∈ absconv ( δ K 2 n +1 ) l X i =1 | α ( u i ) | T o obtain (4.33) we can put λ = card F . Pr o of of The or em 4.10. Consider the s equence o f r ectangles as in Theore m 4.3: E N = ( f N )  By Theor em 4 .3, their union cov ers the whole space O exp ( G ) O exp ( G ) = ∞ [ N =1 E N and moreover any dua lly submultiplicativ e rectangle D ⊆ O ( G ) is contained in so me rectang le E N : D ⊆ E N F r om this we deduce tha t in F ormulas (4 .16) and (4.17) the system o f all dually submult iplicative rect- angles D ca n b e replaced by the sys tem of rec tangles E N : O exp ( G ) = ∞ [ N =1 C E N = lim → N →∞ C E N This means that O exp ( G ) is a B rauner space, a nd O ⋆ exp ( G ) a F r´ echet s pace. T o prov e that b oth spaces are nuclear we can only note that O exp ( G ) is c on uclear. Co nsider C E N as Ba nac h spa ces with unit balls E N . B y (4.14), the seminorm on C E N is deﬁned by equality p N ( u ) = sup n ∈ N 1 N n | u | K n § 4. FUNCTIONS OF EXPONENTIAL TYPE ON A STEIN GROUP 79 Its unit ball is the se t E N : E N = { u ∈ O ( G ) : ∀ n ∈ N | u | K n 6 N n } =  u ∈ O ( G ) : ∀ n ∈ N 1 N n | u | K n 6 1  = =  u ∈ O ( G ) : p N ( u ) = sup n ∈ N 1 N n | u | K n 6 1  T o prov e that the s pace O exp ( G ) = lim → N →∞ C E N is conuclear, we need to verify that for any N ∈ N there exists M ∈ N , M > N , such that the inclusion ma pping C E N → C E M is a bsolutely summing, i.e. suc h that for some constant L > 0 , for any l ∈ N and for all u 1 , ..., u l ∈ C E N we hav e: l X i =1 p M ( u i ) 6 L · sup α ∈ ( E N ) ◦ l X i =1 | α ( u i ) | (4.38) This is prov ed a s follows. Firs t we need to note that for all n, N ∈ N the following inclusion holds E N = { u ∈ O ( G ) : ∀ n ∈ N | u | K n 6 N n } ⊆ { u ∈ O ( G ) : | u | K n 6 N n } = N n · ◦ δ K n (4.39) It implies following chain: E N ⊆ N n · ◦  δ K n  ⇓ ( E N ) ◦ ⊇ ( N n · ◦  δ K n  ) ◦ = 1 N n · ( ◦  δ K n  ) ◦ = 1 N n · absconv ( δ K n ) ⇓ absconv ( δ K n ) ⊆ N n · ( E N ) ◦ ⇓ absconv ( δ K 2 n +1 ) ⊆ N 2 n +1 · ( E N ) ◦ ⇓ l X i =1 | u i | K n 6 (4.33) 6 C · λ n − 1 · sup α ∈ absconv ( δ K 2 n +1 ) l X i =1 | α ( u i ) | 6 C · λ n − 1 · sup α ∈ N 2 n +1 · ( E N ) ◦ l X i =1 | α ( u i ) | = = C · λ n − 1 · sup β ∈ ( E N ) ◦ l X i =1 | N 2 n +1 · β ( u i ) | = C · λ n − 1 · N 2 n +1 · sup β ∈ ( E N ) ◦ l X i =1 | β ( u i ) | ⇓ ∀ M > 0 l X i =1 1 M n | u i | K n 6 C · λ n − 1 · N 2 n +1 M n · sup β ∈ ( E N ) ◦ l X i =1 | β ( u i ) | ⇓ l X i =1 p M ( u i ) = l X i =1 sup n ∈ N 1 M n | u i | K n 6 l X i =1 ∞ X n =1 1 M n | u i | K n ! = ∞ X n =1 l X i =1 1 M n | u i | K n ! 6 6 ∞ X n =1 C · λ n − 1 · N 2 n +1 M n · sup β ∈ ( E N ) ◦ l X i =1 | β ( u i ) | ! = C · ∞ X n =1 λ n − 1 · N 2 n +1 M n ! · sup β ∈ ( E N ) ◦ l X i =1 | β ( u i ) | 80 If we now cho ose M ∈ N s uﬃcien tly large such that C · ∞ X n =1 λ n − 1 · N 2 n +1 M n 6 1 then the constant L in (4.3 8) ca n b e chosen as 1: l X i =1 p M ( u i ) 6 C · ∞ X n =1 λ n − 1 · N 2 n +1 M n | {z } 6 1 · sup β ∈ ( E N ) ◦ l X i =1 | β ( u i ) | (g) Holomorphic mappings of exp onen tial type and tensor pro ducts of the spaces O exp ( G ) and O ⋆ exp ( G ) Theorem 4.11. L et G and H b e two c omp actly gener ate d Stein gr oups. The formula ρ G,H ( u ⊡ v ) = u ⊙ v , u ∈ O exp ( G ) , v ∈ O exp ( H ) (4.40) (wher e u ⊡ v is the function fr om (1.24) ) deﬁnes a line ar c ontinuous mapping ρ G,H : O exp ( G × H ) → O exp ( G ) ⊘ O ⋆ exp ( H ) = O exp ( G ) ⊙ O exp ( H ) , This mapping is an isomorphism of ster e otyp e sp ac es and is natur al by G and H , i.e. is an isomorphism of bifunctors fr om t he c ate gory SG of St ein gr oups int o the c ate gory Ste of ster e otyp e sp ac es:  ( G ; H ) 7→ O ( G × H )  ֌  ( G ; H ) 7→ O ( G ) ⊙ O ( H )  Equivalently this mapping is deﬁne d by formula ρ G,H ( w )( β ) = β ◦ b w , w ∈ O exp ( G × H ) , β ∈ O ⋆ exp ( H ) (4.41) wher e b w : G → O exp ( H )    b w ( s )( t ) = w ( s, t ) , s ∈ G, t ∈ H (4.42) Corollary 4. 4. The fol lowing isomorphisms of funct ors hold: O exp ( G × H ) ∼ = O exp ( G ) ⊙ O exp ( H ) ∼ = O exp ( G ) ⊛ O exp ( H ) (4.43) O ⋆ exp ( G × H ) ∼ = O ⋆ exp ( G ) ⊙ O ⋆ exp ( H ) ∼ = O ⋆ exp ( G ) ⊛ O ⋆ exp ( H ) (4.44) T o prov e Theorem 4 .11 we hav e to r ecall the no tion of injective tensor pro duct A ⊙ B of sets A and B in ster eot yp e spaces X and Y . According to notations [1, (7.27 )], A ⊙ B is deﬁned as a subset in the space X ⊙ Y = X ⊘ Y ⋆ of op erators ϕ : Y ⋆ → X co n taining only those op erators sa tisfying the condition ϕ ( B ◦ ) ⊆ A . That is the sens e of the following fo rm ula we use in Lemma 4.5 b elow: A ⊙ B = A ⊘ B ◦ . Lemma 4. 5. If g : G → R + and h : H → R + ar e two semichar acters on G and H , then g ⊡ h is a semichar acter on G × H , and the mappi ng ρ G,H deﬁne d in (4.41) - (4 .42) is a home omorphism b etwe en c omp act set s ( g ⊡ h )  ⊆ O exp ( G × H ) and g  ⊙ h  ⊆ O exp ( G ) ⊙ O exp ( H ) : ( g ⊡ h )  ∼ = g  ⊙ h  (4.45) § 4. FUNCTIONS OF EXPONENTIAL TYPE ON A STEIN GROUP 81 Pr o of. Here at the be ginning we use reasoning s similar to those we us ed in the pr oof of Theo rem 4.8. 1. Note ﬁrst that for a n y function w ∈ ( g ⊡ h )  ⊆ O exp ( G × H ) F ormula (4 .42) deﬁnes s ome mapping b w : G → O exp ( H ) Indeed, s ince w is holo morphic on G × H , it is holomo rphic with resp ect to each of t wo v ariables, s o when s ∈ G is ﬁxed, then the function b w ( s ) : H → C is als o holomorphic. At the s ame time it is bo unded by the semicharacter g ( s ) · h : ∀ s ∈ G b w ( s ) ∈ g ( s ) · h  (4.46) since w ∈ ( g ⊡ h )  = ⇒ | b w ( s )( t ) | = | w ( s, t ) | 6 g ( s ) · h ( t ) = ⇒ b w ( s ) ∈ g ( s ) · h  Thu s, b w ( s ) is alwa ys a ho lomorphic function of exp onen tial type on H , i.e. b w (( s ) ∈ O exp ( H ) 2. Let us show that the mapping b w : G → O exp ( H ) is co n tinuous. Let s i be a sequence of p oints in G tending to a p oint s : s i G − → i →∞ s Then the sequence of holomor phic functions b w ( s i ) ∈ O ( H ) tends to the holomor phic function b w ( s ) ∈ O ( H ) unifor mly on ea c h co mpact set K ⊆ H , i.e. in the space O ( H ): b w ( s i ) O ( H ) − → i →∞ b w ( s ) On the other hand, all functions are b ounded b y the semich ar acter C · h , wher e C = max { sup i g ( s i ) , g ( s ) } is a ﬁnite num ber , since the sequence s i conv erges and together with its limit is a co mpact set: | b w ( s i )( t ) | 6 g ( s i ) · h ( t ) 6 C · h ( t ) Thu s, the functions b w ( s i ) and b w ( s ) lie in a rectangle ge nerated by the semicharacter C · h : { b w ( s i ); b w ( s ) } ⊆ ( C · h )  In other words, b w ( s i ) tends to b w ( s ) o n the compac t set ( C · h )  b w ( s i ) ( C · h )  − → i →∞ b w ( s ) so b w ( s i ) tends to b w ( s ) in the space O exp ( H ): b w ( s i ) O exp ( H ) − → i →∞ b w ( s ) 3. F r om contin uity of the mapping b w : G → O exp ( H ) it follows that for any functional β ∈ O ⋆ exp ( H ) the function β ◦ b w : G → C is ho lomorphic. T o prov e this one ca n use the Mor era theorem: consider a closed oriented hypers urface Γ in G o f dimension n = dim G with a s uﬃcien tly little diameter and show that Z Γ ( β ◦ b w )( s ) d s = 0 (4.47) Indeed, take a net of functionals { β i ; i → ∞} ⊂ O ⋆ exp ( H ) which are linea r co m binations of delta- functionals, and approximate β in O ⋆ exp ( H ): β i = X k λ k i · δ a k i , β i O ⋆ exp ( H ) − → i →∞ β Then we obtain the fo llo wing. Since b w : G → O exp ( H ) is contin uous, we hav e β ◦ b w C ( G ) ← − ∞← i β i ◦ b w 82 This implies that for any Radon measure α ∈ C ( G ) α ( β ◦ b w ) ← − ∞← i α ( β i ◦ b w ) In particular , for the functional of integrating by our hypersurfa ce Γ w e obtain Z Γ ( β ◦ b w )( s ) d s ← − ∞← i Z Γ ( β i ◦ b w )( s ) d s = Z Γ X k λ k i · δ a k i ◦ b w ! ( s ) d s = = X k λ k i · Z Γ  δ a k i ◦ b w  ( s ) d s = X k λ k i · Z Γ δ a k i ( b w ( s )) d s = = X k λ k i · Z Γ b w ( s )( a k i ) d s = X k λ k i · Z Γ w ( s, a k i ) d s | {z } k 0 , since w is holomorp hic with re spec t to the ﬁrst v ariable = 0 I.e., indeed (4.47) is true. 4. W e understo o d that for any functiona l β ∈ O ⋆ exp ( H ) the function β ◦ b w : G → C is holomor phic. Let us show now that it is of exp onen tial type: ∀ w ∈ O exp ( G × H ) ∀ β ∈ O ⋆ exp ( H ) β ◦ b w ∈ O exp ( G ) (4.48) Indeed, since the functional β ∈ O ⋆ exp ( H ) is bo unded on the compact se t h  ⊆ O exp ( H ), it must b e a bo unded functiona l o n the Ba nac h representation of the Smith space C h  , i.e. ∀ v ∈ C h  | β ( v ) | 6 M · k v k h  (4.49) where M = k β k ( h  ) ◦ := max v ∈ h  | β ( v ) | , k v k h  := inf { λ > 0 : v ∈ λ · h  } So from formula (4.4 6 ) we hav e: b w ( s ) ∈ g ( s ) · h  = ⇒ k b w ( s ) k h  := inf { λ > 0 : b w ( s ) ∈ λ · h  } 6 g ( s ) = ⇒ = ⇒ | β ( b w ( s )) | 6 M · g ( s ) I.e. the function β ◦ b w is bo unded by the semicharacter M · g : β ◦ b w ∈ M · g  (4.50) 5. W e hav e prov ed (4 .48). Now let us show that for any w ∈ ( g ⊡ h )  ⊆ O exp ( G × H ) the mapping β ∈ O ⋆ exp ( H ) 7→ ρ G,H ( w )( β ) = β ◦ b w ∈ O exp ( G ) (4.51) is contin uous, i.e. ρ G,H ( w ) ∈ O exp ( G ) ⊘ O ⋆ exp ( H ) (4.52) This follows fro m (4.50): if β i is a net tending to zer o in O ⋆ exp ( H ), then β i ◦ b w ∈ M i · g  , M i = max v ∈ h  | β i ( v ) | − → i →∞ 0 ⇓ β i ◦ b w g  − → i →∞ 0 § 4. FUNCTIONS OF EXPONENTIAL TYPE ON A STEIN GROUP 83 ⇓ β i ◦ b w O exp ( G ) − → i →∞ 0 6. Now we need to verify that ρ G,H ( w ) ∈ g  ⊙ h  = g  ⊘ ( h  ) ◦ ⊆ O exp ( G ) ⊘ O ⋆ exp ( H ) (4.53) Or, in other words, ρ G,H ( w )  ( h  ) ◦  ⊆ g  This follows fro m (4.46): ∀ s ∈ G b w ( s ) ∈ g ( s ) · h  ⇓ ∀ s ∈ G 1 g ( s ) b w ( s ) ∈ h  ⇓ ∀ s ∈ G ∀ β ∈ ( h  ) ◦ 1 >     β  1 g ( s ) b w ( s )      = 1 g ( s ) | ( β ◦ b w )( s ) | = 1 g ( s ) | ( ρ G,H ( w )( β ))( s ) | ⇓ ∀ s ∈ G ∀ β ∈ ( h  ) ◦ | ρ G,H ( w )( β )( s ) | 6 g ( s ) ⇓ ∀ β ∈ ( h  ) ◦ ρ G,H ( w )( β ) ∈ g  ⇓ ρ G,H ( w )  ( h  ) ◦  ⊆ g  7. Let us show that the mapping w ∈ ( g ⊡ h )  7→ ρ G,H ( w ) ∈ g  ⊙ h  (4.54) is injective. Consider functionals o f the fo rm δ s,t : O exp ( G ) ⊘ O ⋆ exp ( H ) → C    δ s,t ( ϕ ) = ϕ ( δ t )( s ) ( 4.55 ) Now we hav e: if w 6 = 0, then for some s ∈ G , t ∈ H we hav e w ( s, t ) 6 = 0, so δ s,t ( ρ G,H ( w )) = ρ G,H ( w )( δ t )( s ) = ( δ t ◦ b w )( s ) = δ t  b w ( s )  = b w ( s )( t ) = w ( s, t ) 6 = 0 th us, ρ G,H ( w ) 6 = 0. 8. Similar ly it turns out that the mapping w ∈ ( g ⊡ h )  7→ ρ G,H ( w ) ∈ g  ⊙ h  (4.56) is surjective: for any ϕ ∈ g  ⊙ h  = g  ⊘ ( h  ) ◦ ⊂ O exp ( G ) ⊘ O ⋆ exp ( H ) we set w ( s, t ) = δ s ⊛ δ t ( ϕ ) = ϕ ( δ t )( s ) , s ∈ G, t ∈ H and then, ﬁrs t, w is a ho lomorphic function on G × H , since it is holomor phic with resp ect to every v ar iable: when t ∈ H is ﬁxed, the ob ject ϕ ( δ t ) is an element of the s pace O exp ( G ), i.e. a holomorphic function (of exp onential type ) on G , hence w ( · , t ) is holo morphic with r espect to the ﬁr st v a riable, and 84 when s ∈ G is ﬁxed, the mapping β ∈ O ⋆ exp ( H ) 7→ ( δ s ◦ ϕ )( β ) is a contin uous functional o n the space O ⋆ exp ( H ), i.e. by ster eot yp e duality , a n element o f the space O exp ( H ): ( δ s ◦ ϕ )( β ) = β ( v ) , v ∈ O exp ( H ) Thu s w ( s, t ) = ϕ ( δ t )( s ) = ( δ s ◦ ϕ )( δ t ) = δ t ( v ) = v ( t ) i.e. the function w ( s, · ) is holomor phic with resp ect to the second v ariable. F ur ther, turning the chain of item 6 to reverse direction, we obtain: ϕ ∈ g  ⊙ h  ⇓ ϕ  ( h  ) ◦  ⊆ g  ⇓ ∀ β ∈ ( h  ) ◦ ϕ ( β ) ∈ g  ⇓ ∀ t ∈ H 1 h ( t ) δ t ∈ ( h  ) ◦ = ⇒ ϕ  1 h ( t ) δ t  ∈ g  ⇓ ∀ t ∈ H ϕ ( δ t ) ∈ h ( t ) · g  ⇓ ∀ s ∈ G ∀ t ∈ H | w ( s, t ) | = | ϕ ( δ t )( s ) | 6 h ( t ) · g ( s ) = | ( g ⊡ h )( s, t ) | ⇓ w ∈ ( g ⊡ h )  And ﬁnally it re mains to note tha t w is an inv erse imag e of ϕ under the mapping w 7→ ρ G,H ( w ): ∀ s, t ρ G,H ( w )( δ t )( s ) = ( δ t ◦ b w )( s ) = δ t ( b w ( s )) = b w ( s )( t ) = w ( s, t ) = ϕ ( δ t )( s ) ⇓ ρ G,H ( w ) = ϕ 9. Thus, we obtained that the mapping w ∈ ( g ⊡ h )  7→ ρ G,H ( w ) ∈ g  ⊙ h  (4.57) is bijective. It r emains to s ho w now that it is contin uous in bo th direc tions. This fo llo ws from the fact that both ( g ⊡ h )  and g  ⊙ h  are compact sets. Since functionals δ s ⊛ δ t separate the po in ts of the compact set g  ⊙ h  , the (Hausdo rﬀ ) topo logy they generate on g  ⊙ h  coincides with the initial top ology of g  ⊙ h  : if δ s ⊛ δ t ( ϕ i ) − → i →∞ δ s ⊛ δ t ( ϕ ) for any s ∈ G , t ∈ H , then ϕ i g  ⊙ h  − → i →∞ ϕ F r om this w e have that the mapping w 7→ ρ G,H ( w ) is contin uous in forward direction: w i ( g ⊡ h )  − → i →∞ w § 4. FUNCTIONS OF EXPONENTIAL TYPE ON A STEIN GROUP 85 ⇓ ∀ ( s, t ) ∈ G × H δ s ⊛ δ t ( ρ G,H ( w i )) = w i ( s, t ) − → i →∞ w ( s, t ) = δ s ⊛ δ t ( ρ G,H ( w )) ⇓ ρ G,H ( w i ) g  ⊙ h  − → i →∞ ρ G,H ( w ) Thu s, the op eration w 7→ ρ G,H ( w ) is a contin uous bijective mapping of the co mpact set ( g ⊡ h )  int o the compact set g  ⊙ h  . T his means that ρ G,H is a homeomor phism b et ween ( g ⊡ h )  and g  ⊙ h  . Pr o of of The or em 4.11. Let us note from the very b eginning that if f is a s emic harac ter on G × H , then the functions g ( s ) = f ( s, 1 H ) , h ( t ) = f (1 G , t ) , s ∈ G, t ∈ H are again semicharacters (as restrictio ns of f on subgr oups), and the function g ⊡ h is a semicharacter on G × H , ma jorizing f : f 6 g ⊡ h (4.58) Indeed, f ( s, t ) = f (( s, 1 H ) · (1 G , t )) 6 f ( s, 1 H ) · f (1 G , t ) = g ( s ) · h ( t ) = ( g ⊡ h )( s, t ) F r om this it follows that ev ery function w ∈ O exp ( G × H ) is contained in so me compact set of the form ( g ⊡ h )  (since w is alwa ys contained in compact set of the for m f  ). At the same time the ob ject ρ G,H ( w ) is an element of the set g  ⊙ h  , i.e. an element of the space O exp ( G ) ⊘ O ⋆ exp ( H ). Thu s, F ormulas (4.4 2 ) and (4.4 1) co rrectly deﬁne a ma pping w ∈ O exp ( G × H ) 7→ ρ G,H ( w ) ∈ O exp ( G ) ⊘ O ⋆ exp ( H ) (4.59) and we only need to chec k its bijectivity and co n tinuit y in b oth directio ns. 1. The injectivit y of ρ G,H follows from its injectivit y on compact se ts ( g ⊡ h )  (and from the fact that the compact sets ( g ⊡ h )  and g  ⊙ h  are injectively included into the s paces O exp ( G × H ) and O exp ( G ) ⊘ O ⋆ exp ( H )). 2. The surjectivity of ρ G,H follows from the fact that it sur jectiv ely maps compact sets ( g ⊡ h )  int o compact sets g  ⊙ h  (and fro m the fact that the co mpact sets g  ⊙ h  cov er all the space O exp ( G ) ⊘O ⋆ exp ( H )). 3. The contin uity follows fro m the fact tha t ρ G,H contin uously maps every compact set ( g ⊡ h )  int o the co mpact set g  ⊙ h  , henc e int o the space O exp ( G ) ⊘ O ⋆ exp ( H ). This mea ns that ρ G,H is contin uous on each compact set K in the Brauner space O exp ( G × H ), thus it must b e contin uous on the whole space O exp ( G × H ). 4. The contin uity in reverse direction is proved in the s ame way: since the inv erse mapping contin u- ously turns every co mpact set g  ⊙ h  int o a co mpact set ( g ⊡ h )  , it is contin uous on each co mpact s et in the Brauner space O exp ( G ) ⊘ O ⋆ exp ( H ). Thus, it is contin uous on the whole space O exp ( G ) ⊘ O ⋆ exp ( H ). 5. W e hav e proved that the ma pping deﬁned by formulas (4.41) - (4.42) is an isomo rphism of the stereotype spaces : O exp ( G × H ) ∼ = O exp ( G ) ⊘ O ⋆ exp ( H ) = O exp ( G ) ⊙ O exp ( H ) (hence the identities (4.43) hold). Let us show that this mapping satisﬁes the identit y (4.40): if u ∈ O exp ( G ), v ∈ O exp ( H ), then for the mapping [ u ⊡ v : G → O exp ( H ) deﬁned by for m ula (4.4 2) we hav e the following log ic chain: [ u ⊡ v ( s )( t ) = ( u ⊡ v )( s, t ) = u ( s ) · v ( t ) ⇓ [ u ⊡ v ( s ) = u ( s ) · v ⇓ ∀ β ∈ O ⋆ exp ( H ) β ( [ u ⊡ v ( s )) = u ( s ) · β ( v ) 86 ⇓ ∀ β ∈ O ⋆ exp ( H ) ρ G,H ( u ⊡ v )( β ) = β ◦ [ u ⊡ v = β ( v ) · u = (1.2) = ( u ⊙ v )( β ) ⇓ ρ G,H ( u ⊡ v ) = u ⊙ v It r emains now to note that since the elements of the form u ⊛ v g enerate a dense subs pace in O exp ( G ) ⊛ O exp ( H ), by the pr o ven ab ov e nuclearity of the spaces O exp , the corres ponding elements of the form u ⊙ v m ust g enerate a dense subspace in O exp ( G ) ⊙ O exp ( H ), a nd elements u ⊡ v a dense subspace in O exp ( G × H ). This implies that the pr operty (4 .40) uniquely deﬁne the mapping ρ G,H . (h) Structure of Hopf algebras on O exp ( G ) and O ⋆ exp ( G ) In § 3(b) we hav e mentioned the standar d trick, a llo wing to prove that functional algebr as of a given class on groups are Hopf algebra s – a suﬃcie n t conditio n for this is a natural isomo rphism betw een the functional algebr a of the Cartesia n pro duct of groups × and the co rresp onding tenso r pro duct o f their functional algebr as. Theorem 4.11, establishing the na tural iso morphism O exp ( G × H ) ρ G,H ∼ = O exp ( G ) ⊙ O exp ( H ) allows now to make the same conclusion ab out algebra s O exp ( G ): Theorem 4.12. F or any c omp actly gener ate d St ein gr oup G – the sp ac e O exp ( G ) of holomorphic functions of ex p onential typ e on G is a nucle ar Hopf-Br auner algebr a with re sp e ct to the algebr aic op er ations deﬁne d by formulas, analo gous to (3.3) - (3.7) ; – its ster e otyp e dual sp ac e O ⋆ exp ( G ) is a nucle ar Hopf-F r´ echet algebr a with r esp e ct to dual algebr aic op er ations. § 5 Arens-Mic hael en v elop e and holomorphic reﬂexivit y (a) Subm ult iplicative seminorms and Arens-Mic hael algebras A se minorm p : A → R + on an algebra A is said to b e submultiplic ative , if it satisﬁes the fo llo wing condition p ( u · v ) 6 p ( u ) · p ( v ) , u, v ∈ A This is equiv alen t to the fact tha t the unit ball of this seminor m U = { u ∈ A : p ( u ) 6 1 } satisﬁes the condition U · U ⊆ U (such s ets in A are a lso s aid to b e submu lt ipli c ative like s ubm ultiplicative neighborho o ds of zero, deﬁned on page 66). A top ological a lgebra A is c alled an Ar ens-Michael algebr a , if it is complete (a s a top olog ical vector space) and satisﬁes the following equiv a len t conditions: (i) the top ology of A is genera ted by a system of submultiplicative seminor ms: (ii) A has a lo cal base of submultiplicativ e closed absolutely co n vex neig h b orho ods of zero . § 5. ARENS-MICHAEL E NVELOPE AND HOLO MORPHIC RE FLEXIVITY 87 Example 5.1 . Seminor ms (3.20) ge nerating the top ology on O ( Z ) are submultiplicativ e: || u || N = X | n | 6 N | u ( n ) | , N ∈ N , hence O ( Z ) is an Arens-Michael alg ebra. Pr o of. Indeed, k u · v k N = X | n | 6 N | ( u · v )( n ) | = (3.19) = X | n | 6 N | u ( n ) · v ( n ) | 6   X | n | 6 N | u ( n ) |   ·   X | n | 6 N | v ( n ) |   = k u k N · k v k N Example 5.2 . Seminor ms (3.47) ge nerating the top ology on O ( C × ), are submultiplicativ e: || u || C = X n ∈ Z | u n | · C | n | , C > 1 , hence O ( C × ) is an Arens-Michael alg ebra. Pr o of. Indeed, || u · v || C = X n ∈ Z | ( u · v ) n | · C | n | = (3.46) = X n ∈ Z      X i ∈ Z u i · v n − i      · C | n | 6 6 X n ∈ Z X i ∈ Z | u i | · | v n − i | · C | i | · C | n − i | = X k ∈ Z | u k | · C | k | ! · X l ∈ Z | u l | · C | l | ! = || u || C · || v || C Example 5.3 . Seminor ms (3.72), ge nerating the top ology on O ( C ), are submultiplicativ e: || u || C = ∞ X n =0 | u n | · C n , C > 0 , hence O ( C ) is an Arens-Michael a lgebra. Pr o of. Indeed, || u · v || C = ∞ X n =0 | ( u · v ) n | · C n = (3.71) = ∞ X n =0      n X i =0 u i · v n − i      · C n 6 ∞ X n =0 n X i =0 | u i | · | v n − i | · C n = = X k ∈ N | u k | · C k ! · ∞ X l =0 | u l | · C l ! = || u || C · || v || C Of the following three pro positio ns the ﬁr st tw o ar e evident, and the third one follo ws from the A. Y u. Pirkovskii theore m 5.1 which we formulate b elow in this section: Prop osition 5. 1 . The algebr a O ( M ) of holomorph ic fun ctions on any c omplex manifold M is an Ar ens- Michael algebr a. Prop osition 5. 2 . The alg ebr a O ⋆ exp ( G ) of exp onential fun ct ionals on every Stein gr oup is an Ar ens- Michael algebr a. Prop osition 5.3. The algebr a R ( M ) of p olynomials on a c omplex aﬃne algebr aic manifold M , endowe d with the str ongest lo c al ly c onvex top olo gy, is an A r ens-Michael algebr a if an only if the m anifo ld M is ﬁnite. 88 (b) Arens-Mic hael en v elop es The Ar ens-Michael envelop e of a top ological algebra A is a (contin uous) homomo rphism π : A → B of A int o a n Ar ens-Mich ael algebra B s uc h that for any (co n tinuous) homo morphism ρ : A → C of A into an arbitrar y Arens-Michael algebr a C there is a unique (contin uous) homo morphism σ : B → C such that the following diag ram is commutativ e: A B C / / π   ? ? ? ? ? ρ      σ F r om this deﬁnition it is cle ar that if π : A → B and ρ : A → C are tw o Arens-Michael env elop es of A , then the ar ising ho momorphism σ : B → C beco mes an isomorphism of top ologic al algebra s (due to the uniqueness o f σ ). Hence the Ar ens-Mich ael envelope of A is deﬁned uniquely up to a n is omorphism, and as a corolla ry , we can int ro duce a sp ecial notation for this co nstruction: ♥ A : A → A ♥ This sho uld b e understo o d as follows: if we hav e a ho momorphism ϕ : A → B , then the reco rd ϕ = ♥ A means that ϕ : A → B is an Arens-Michael envelope of the algebra A ; o n the other hand, if we hav e an algebra B , then the r ecord B = A ♥ means that there ex ists a homo morphism ϕ : A → B which is an Arens-Michael env elop e of the a lgebra A — in this cas e the algebra B is a lso called an Ar ens-Michae l envelop e of the algebr a A . Prop osition 5. 4. A top olo gic al algebr a B is an A r ens-Michael envelop e for a top olo gic al algebr a A if and only if (i) B is an Ar ens-Michael algebr a, and (ii) ther e exist s a c ontinuous homomorphi sm π : A → B such that (a) the image π ( A ) of the algebr a A under the action of π is dense in B , (b) for any c ontinuous submultiplic ative seminorm p : A → R + ther e is a c ontinuous submu l- tiplic ative seminorm e p : B → R + , su ch t hat t he seminorm e p ◦ π : A → R + majorizes the seminorm p : p ( a ) 6 e p ( π ( a )) , a ∈ A The Arens-Michael env elop e can b e cons tructed dir ectly: to a n y submultiplicativ e neighbor hoo d o f zero U in A we can ass ign the closed ideal Ker U in A deﬁned by the equality Ker U = \ ε> 0 ε · U and a quotient alge bra A/ Ker U endow ed with the top ology of normed space w ith the unit ball U + Ker U . Then the completion ( A/ K er U ) H bec omes a Banach algebra . F ollowing (0.3) we denote such algebra by A/U : A/U := ( A/ Ker U ) H The family of s uc h algebras (with diﬀeren t U ) f orms a pro jective system. The Its limit is an Arens- Mic hael env elop e A ♥ : A ♥ = lim ← − U is a submultiplicati v e neighborh oo d of ze ro in A A/U. (5.1) The following prop ositions show that the op eration of ta king the Arens -Mic hael env elop e commutes with the passag e to direct sums a nd quotient algebras . § 5. ARENS-MICHAEL E NVELOPE AND HOLO MORPHIC RE FLEXIVITY 89 Prop osition 5. 5 . The Ar ens-Michae l envelop e of a dir e ct sum A 1 ⊕ ... ⊕ A n of ﬁnite family of top olo gic al algebr as A 1 , ..., A n c oincides with the dir e ct su m of the Ar ens-Michael envelop es of t hese algebr as: ( A 1 ⊕ ... ⊕ A n ) ♥ ∼ = A ♥ 1 ⊕ ... ⊕ A ♥ n Prop osition 5. 6. L et π : A → A ♥ b e an Ar ens- Mich ael envelop e of the algebr a A , and let I b e a close d ide al in A . Then the Ar en s - Mich ael envelop e of t he qu otient algebr a A/ I c oincides with the c ompletion of the quotient algebr a A ♥ / π ( I ) over t he closur e π ( I ) in A ♥ of the image of ide al I under the action of the mapping π : ( A/I ) ♥ ∼ =  A ♥ / π ( I )  H An impo rtant ex ample of the Arens-Michael env elop e was constructed by A .Y u. Pirko vskii: Theorem 5.1 (A .Y u. Pir k ovskii, [26]) . The Ar ens-Michael envelop e of the algebr a R ( M ) of p olynomials on an aﬃne algebr aic manifold M c oincides with the algebr a O ( M ) of holomorphic funct ions on M :  R ( M )  ♥ ∼ = O ( M ) (5.2) (c) The mapping ♭ ⋆ G : O ⋆ ( G ) → O ⋆ exp ( G ) is an Arens-Mic hael en v elop e Theorem 5.2. F or any Stein gr oup G the mapping ♭ ⋆ G : O ⋆ ( G ) → O ⋆ exp ( G ) is an Ar ens-Michael envelop e of the algebr a O ⋆ ( G ) :  O ⋆ ( G )  ♥ ∼ = O ⋆ exp ( G ) (5.3 ) Pr o of. F r om the representation (4.17) it fo llo ws that this space is a pr o jective limit of the Banach quotient algebras : O ⋆ exp ( G ) = (4.17) = lim − → D is a dually submultiplicative rectangle in O ( G ) C D ! ⋆ = lim ← − D i s a dually submultiplicative rectangle in O ( G ) ( C D ) ⋆ = (0.4) = lim ← − D i s a dually submultiplicative rectangle in O ( G ) O ⋆ ( G ) /D ◦ = = lim ← − ∆ i s a submultiplicative rhombus in O ⋆ ( G ) O ⋆ ( G ) / ∆ = (Theor em 4.2(a)) = lim ← − U is a submultiplicati v e neighborh oo d of ze ro in O ⋆ ( G ) O ⋆ ( G ) /U = (5.1) =  O ⋆ ( G )  ♥ (d) The mapping ♭ G : O exp ( G ) → O ( G ) is an Arens-Mic hael en v elop e for the groups with algebraic connected c omp onen t of iden tity Theorem 5. 3. L et G b e a c omp actly gener ate d S tein gr oup, such that the c onne cte d c omp onent of identity G e is an algebr aic gr oup. Then the mapping ♭ G : O exp ( G ) → O ( G ) is an Ar ens-Michael envelop e of the algebr a O exp ( G ) :  O exp ( G )  ♥ ∼ = O ( G ) (5.4) 90 W e shall prov e this theorem in several steps . 1. Let initially G b e a disc rete group. Reca ll that in (0.28) we agr eed to denote by 1 x the characteristic functions of single tons { x } in G : 1 x ( y ) = ( 1 , y = x 0 y 6 = x (5.5) (since G is disc rete, the function 1 x can be considered as element of bo th a lgebras O ( G ) and O exp ( G )). Lemma 5.1. The functions { 1 x ; x ∈ G } form a b asis in the t op olo gic al ve ctor sp ac es O ( G ) and O exp ( G ) : for any function u ∈ O ( G ) ( u ∈ O exp ( G ) ) t he fol lowing e qu ality holds u = X x ∈ G u ( x ) · 1 x (5.6) wher e the s eries c onver ges in O ( G ) ( O exp ( G ) ), and its c o eﬃcients c ontinuously dep en d on u ∈ O ( G ) ( u ∈ O exp ( G ) ). Pr o of. F o r the space O ( G ) this is evident, since for the case of disc rete group G this s pace coincides with the space C G of all functions o n G . Let us prove this for O exp ( G ): if u ∈ O exp ( G ), then taking a ma jorizing semicharacter f : G → R + , | u ( x ) | 6 f ( x ) , x ∈ G we o btain that the partia l sums of the series (5.6 ) ar e co n tained in the rectangle f  , so the ser ies (5.6 ) conv erges (not only in O ( G ), but) in O exp ( G ) a s well. On the other ha nd, every co eﬃcien t u ( x ) contin- uously depe nd u , if u r uns ov er the rectang le f  . B y the deﬁnition of top ology in O exp ( G ), this mea ns that u ( x ) contin uously dep end on u , when u runs over O exp ( G ). Lemma 5.2. If G is a discr ete ﬁnitely gener ate d gr oup, then for any c ontinuous seminorm q : O exp ( G ) → R + and for any semichar acter f : G → [1 ; + ∞ ) the nu mb er family { f ( x ) · q (1 x ); x ∈ G } is summable: X x ∈ G f ( x ) · q (1 x ) < ∞ Pr o of. Let T is an absolutely convex co mpact set in O ⋆ exp ( G ) corr espo nding to the seminorm q : q ( u ) = sup α ∈ T | α ( u ) | Every rectang le f  is a compact set in O ⋆ exp ( G ), so ∞ > sup u ∈ f  sup α ∈ T | α ( u ) | = (5.6) = sup u ∈ f  sup α ∈ T    α  X x ∈ G u ( x ) · 1 x     = sup u ∈ f  sup α ∈ T    X x ∈ G u ( x ) · α (1 x )    > > sup α ∈ T    X x ∈ G f ( x ) · α (1 x ) | α (1 x ) | | {z } ↑ one of th e v alues of u ∈ f  · α (1 x )    = sup α ∈ T X x ∈ G f ( x ) · | α (1 x ) | > sup α ∈ T sup x ∈ G f ( x ) · | α (1 x ) | = = sup x ∈ G f ( x ) · sup α ∈ T | α (1 x ) | = sup x ∈ G f ( x ) · q (1 x ) So we hav e tha t for a n y semicharacter f : G → [1; + ∞ ) sup x ∈ G f ( x ) · q (1 x ) < ∞ § 5. ARENS-MICHAEL E NVELOPE AND HOLO MORPHIC RE FLEXIVITY 91 Now take a ﬁnite s et K , g enerating G , ∞ [ n =1 K n = G and deﬁne a semicharacter g : G → [1; + ∞ ) by formula g ( x ) = R n ⇐ ⇒ x ∈ K n \ K n − 1 where R is a num b er, greater than the cardina lit y of K : R > card K. Since the pro duct g · f is a lso a semicharacter, we have: sup x ∈ G h g ( x ) · f ( x ) · q (1 x ) i < ∞ ⇓ ∃ C > 0 ∀ x ∈ G f ( x ) · q (1 x ) 6 C g ( x ) ⇓ X x ∈ G f ( x ) · q (1 x ) 6 X x ∈ G C g ( x ) = ∞ X n =1 X x ∈ K n \ K n − 1 C g ( x ) = ∞ X n =1 X x ∈ K n \ K n − 1 C R n 6 ∞ X n =1 C · ca rd ( K n ) R n 6 6 ∞ X n =1 C · ( ca rd K ) n R n = C · ∞ X n =1  card K R  n < ∞ If q : O exp ( G ) → R + is a contin uous se minorm on O exp ( G ), then let us call its supp ort the s et supp ( q ) = { x ∈ G : q (1 x ) 6 = 0 } (5.7) Lemma 5. 3. If G is a discr ete ﬁ nitely gener ate d gr oup, then for any submu ltiplic ative c ontinuous semi- norm q : O exp ( G ) → R + (a) its supp ort su p p ( q ) is a ﬁnite set: card supp ( q ) < ∞ (b) for any p oint x ∈ supp ( q ) the value of the seminorm q on any function 1 x is less than 1: q (1 x ) > 1 Pr o of. Let us prov e (b) ﬁr st. If x ∈ s upp ( q ), i.e. q (1 x ) > 0, then: 1 x = 1 2 x = ⇒ q (1 x ) = q (1 2 x ) 6 q (1 x ) 2 = ⇒ 1 6 q (1 x ) Now (a). Since the consta n t identit y f ( x ) = 1 is a semicharacter on G , by Lemma 5.2 the num ber family { q (1 x ); x ∈ G } is summing: X x ∈ G q (1 x ) < ∞ On the other hand, by the condition (b) we have alre ady pr o ved, all non-ze ro terms in this series are bo unded fr om below b y 1. Hence there is a ﬁnite num b er of them. 92 2. Let us now pass to the ca se when G is compactly g enerated Stein g roup, who se connected comp onent of identit y G e is a n algebra ic group. Let L C exp ( G ) denote a subalgebr a in O exp ( G ) consisting of lo cally constant functions: u ∈ L C exp ( G ) ⇐ ⇒ u ∈ O exp ( G ) &  ∀ x ∈ G ∃ neighbor hoo d U ∋ x ∀ y ∈ U u ( x ) = u ( y )  Lemma 5.4. L et π : G → G/G e denote the quotient mapping. F or any function v ∈ O exp ( G/G e ) the c omp osition v ◦ π is a lo c al ly c onstant function of exp onential t yp e on G , and t he mapping v 7→ v ◦ π establishes an isomorphism of t op olo gic al algebr as: O exp ( G/G e ) ∼ = L C exp ( G ) Let now for an y coset K ∈ G/G e and for any function u ∈ O exp ( G ) the sym b ol u K denote the function coinciding with u o n the set K ⊂ G a nd v anishing outside of K : u K ( x ) = ( u ( x ) , x ∈ K 0 , x / ∈ K (5.8) The following pro pos ition is proved just like Lemma 5.1 : Lemma 5.5. F or any c oset K ∈ G/G e and for any function u ∈ O exp ( G ) (a) the function u K b elongs t o algebr a O exp ( G ) , (b) the mapping u ∈ O exp ( G ) 7→ u K ∈ O exp ( G ) is c ontinuous, and (c) the series P K ∈ G/G e u K c onver ges in the sp ac e O exp ( G ) t o the function u : u = X K ∈ G/G e u K F o r a n y coset K ∈ G/G e let us consider the op erator of pro jection P K : O exp ( G ) → O exp ( G ) , P K ( u ) = u K and let O exp ( K ) denote its image in the spac e O exp ( G ): O exp ( K ) = P K  O exp ( G )  Clearly , O exp ( K ) is a closed subs pace in O exp ( G ), so O exp ( K ) can b e endowed with the top ology induced from O exp ( G ) (this will b e the same as the top ology of an immediate s ubspace in O exp ( G )), and with resp ect to this top ology O exp ( K ) is a Br auner space. Let in a ddition O ( K ) deno te the us ual algebra of holo morphic functions on the complex manifold K . Lemma 5.6. The inclusion O exp ( K ) ⊆ O ( K ) is an Ar ens-Michael envelop e: O exp ( K ) ♥ = O ( K ) Pr o of. W e hav e to note ﬁrst that it is suﬃcien t to consider the case of K = G e , since shifts turn inclusions O exp ( K ) ⊆ O ( K ) into inclusions O exp ( G e ) ⊆ O ( G e ). F or this case o ur prop osition b ecomes a co rollary of the Pirkovskii Theorem 5 .1: by ass umption, G e is an algebr aic group, so we can cons ider the alg ebra § 5. ARENS-MICHAEL E NVELOPE AND HOLO MORPHIC RE FLEXIVITY 93 R ( G e ) of p olynomials on G e . Then the tra in of thought is illustra ted by the following diag ram (wher e the horizontal arr o ws mean inclusions): R ( G e ) O exp ( G e ) O ( G e ) B / / $ $ J J J J J J J J J J J ρ R / /   ρ z z t t t t t t e ρ If ρ : O exp ( G e ) → B is a n a rbitrary mo rphism into the Arens-Michael a lgebra in B , then by Pirkovskii’s Theorem there a rises a unique morphism ρ R : R ( G e ) → B . Since R ( G e ) is dense in O exp ( G e ) in the top ology o f O ( G e ), the morphism e ρ e xtends the mo rphism ρ . And since O exp ( G e ) is dens e in O ( G e ), this extension is unique. Pr o of of The or em 5.3. Let G be an arbitra ry compactly genera ted Stein gro up with algebra ic co nnected comp onen t of identit y and p : O exp ( G ) → R + a s ubm ultiplicative seminorm. Its res triction p | L C exp ( G ) to the subalgebra L C exp ( G ) deﬁnes by Lemma 5.4 a contin uous seminorm q on O exp ( G/G e ), q ( v ) = p ( v ◦ π ) , and q will b e submultiplicativ e, like p . Hence by Le mma 5.3, the s upport of q is ﬁnite: card supp ( q ) < ∞ Being applied to semino rm p this mean that there exists a ﬁnite family o f cos ets { K 1 , ..., K n } ⊆ G/G e , for which p (1 K i ) 6 = 0 while for the others K ∈ G/G e , K / ∈ { K 1 , ..., K n } , p (1 K ) = 0 (5.9) (here 1 K denotes the image of the co nstan t identit y u ( x ) = 1 under the pro jection (5.8)). As a coro llary , for any function u ∈ O exp ( G ) and for any cos et K / ∈ { K 1 , ..., K n } we hav e p ( u K ) = 0 (5.10) (since p ( u K ) = p (1 K · u K ) 6 p (1 K ) · p ( u K ) = 0 · p ( u K ) = 0). Denote now by P and S the pro jections to the s paces consisting of functions v anishing outside and inside K 1 ∪ ... ∪ K n : P ( u )( x ) =  u ( x ) , x ∈ K 1 ∪ ... ∪ K n 0 , x / ∈ K 1 ∪ ... ∪ K n  = u K 1 + ... + u K n S ( u )( x ) =  u ( x ) , x / ∈ K 1 ∪ ... ∪ K n 0 , x ∈ K 1 ∪ ... ∪ K n  = X K / ∈{ K 1 ,...,K n } u K (the latter ser ies conv erges in O exp ( G ), by Lemma 5 .5). F rom (5.10) we hav e ∀ u ∈ O exp ( G ) p ( S ( u )) = p   X K / ∈{ K 1 ,...,K n } u K   6 X K / ∈{ K 1 ,...,K n } p ( u K ) = (5.10) = X K / ∈{ K 1 ,...,K n } 0 = 0 , what implies p = p ◦ P (5.11) 94 (on the o ne hand, p ( u ) = p ( P ( u ) + S ( u )) 6 p ( P ( u )) + p ( S ( u )) = p ( P ( u )), and on the o ther hand, p ( P ( u )) = p ( u − S ( u )) 6 p ( u ) + p ( S ( u )) = p ( u )). Denote by p i the (submult iplicative) seminorms on O exp ( K i ), induced by p , p i ( v ) = p ( v ) , v ∈ O exp ( K i ) By Lemma 5.6 and Pr opos ition 5.4, these seminor ms are ma jorized b y s ome seminor ms e p i and O ( K i ): p i ( v ) 6 e p i ( v ) , v ∈ O exp ( K i ) F r om (5.11) w e have the following estimation: p ( u ) = p  n X i =1 u K i + X K / ∈{ K 1 ,...,K n } u K  = (5.11) = p  n X i =1 u K i  6 n X i =1 p i ( u K i ) 6 n X i =1 e p i ( u K i ) Thu s, our initial seminorm p on O exp ( G ) is ma jorized by th e seminorm P n i =1 e p i on O ( G ). Again applying Prop osition 5.4, we obtain that the inclusion O exp ( G ) ⊆ O ( G ) is an Arens- Mic hael envelope. (e) Holomorphic reﬂexivit y W e ca n now declar e the following a result of o ur co nsiderations. F or a compactly genera ted Stein gr oup G with the a lgebraic connected comp onent of identit y t wo algebras of those w e considere d abov e, namely , O ⋆ ( G ) and O exp ( G ), hav e the following curious prop erty: every such algebr a H , being a r igid stereotype Hopf a lgebra, has a n Arens-Michael env elop e H ♥ , which also has a structure of rigid stereotype Hopf algebra, and (i) the natural homo morphism ♥ H : H → H ♥ is a homomor phism of rigid Hopf alg ebras, and (ii) the dual mapping ( ♥ H ) ⋆ : ( H ♥ ) ⋆ → H ⋆ is an Arens-Michael env elop e of the algebra ( H ♥ ) ⋆ : ( ♥ H ) ⋆ = ♥ ( H ♥ ) ⋆ Let us note the following in view of this: Prop osition 5. 7. F or an arbitr ary rigid ster e otyp e Hopf algebr a H t he structu r e of rigid Hopf algebr a on the Ar ens-Michael envelop e H ♥ , s atisfyi ng c onditions ( i) and (ii), if exists, is un ique. Pr o of. Note ﬁr st that (i) and (ii) immediately imply (iii) the mappings ♥ H and ( ♥ H ) ⋆ are bimorphis ms of ster eot yp e spaces (i.e. are injective and hav e dense image in the ra nge). Indeed, the mappings ♥ H : H → H ♥ and ( ♥ H ) ⋆ : ( H ♥ ) ⋆ → H ⋆ are epimor phisms (i.e. hav e dense image), since they a re Arens -Mic hael en velopes . On the other hand they are dual to each o ther, so they m ust b e monomor phisms (i.e. are injective). This implies everything. First, the mult iplication and the unit on H ♥ are deﬁned uniquely by the condition that ♥ H : H → H ♥ is the Ar ens-Mic hael env elop e of H . Consider the dual mapping ( ♥ H ) ⋆ : ( H ♥ ) ⋆ → H ⋆ . Like ♥ H , this must be a homomorphism of Hopf algebr as. Hence, it is a homomor phism of algebras , and at the same time an injectiv e mapping, b y virtue of (iii). This m eans that the mult iplication and the unit in ( H ♥ ) ⋆ are deﬁned uniquely since they are unduced from H ⋆ . Thu s, conditions (i) and (ii) impo se r igid co nditions on multiplication, unit, comultiplication and counit in H ♥ , and allow to deﬁne no more than one structure o f bialgebra on H ♥ . On the other hand, we know that antipo de, if exists is also unique, so the structure o f Hopf algebr a on H ♥ is also unique. § 5. ARENS-MICHAEL E NVELOPE AND HOLO MORPHIC RE FLEXIVITY 95 It is conv enient to sketch out conditions (i) and (ii) by the diagra m, H  ♥ / / H ♥ _ ⋆   _ ⋆ O O H ⋆  ♥ o o ( H ♥ ) ⋆ (5.12) with the following sense: ﬁrst, in the corners of the square there are rigid s tereot yp e Hopf alg ebras, and the horizontal arr o ws (the Arens- Mic hael o pera tions ♥ ) ar e their homomorphisms, and, seco nd, the alternation of the op eratio ns ♥ and ⋆ (no matter which pla ce you b egin with) at the fourth step returns back to the initial Hopf a lgebra (of c ourse, up to an isomorphism of functors). The rigid s tereotype Hopf a lgebras H , s atisfying conditions (i) and (ii), will b e ca lled holomorph ic al ly r eﬂexive , and the diagra m (5.12) for such algebra s the r eﬂexivity diagr am . The justiﬁcation of the ter m “reﬂexivity” in this case is the following: if we denote by some s ym b ol, say b , the co mposition of op erations ♥ and ⋆ , b H := ( H ♥ ) ⋆ and call such ob ject a Hopf algebra holomorphic al ly dual to H , then H b ecomes naturally isomor phic to its second dual Hopf a lgebra: H ∼ = b b H (5.13) This is a corollar y of Pro position (5.7): since for holo morphically r eﬂexive Hopf alg ebras the passage H 7→ H ♥ uniquely deﬁnes the structure of Hopf algebra on H ♥ , the isomorphism of algebr as (( H ♥ ) ⋆ ) ♥ ∼ = H ⋆ , po stulating in axiom (ii), a utomatically must b e an isomor phism o f Hopf algebra s. The pass age to the dual Hopf alg ebras exactly gives (5.1 3). Theorems 5.2 and 5.3 imply: Theorem 5.4. I f G is a Stein gr oup with the algebr aic c onn e cte d c omp onent of identity, t hen the algebr as O ⋆ ( G ) and O exp ( G ) ar e holomorphic al ly r eﬂex ive, and the r eﬂexivity diagr am for t hem has the form: O ⋆ ( G )  ♥ (5.3) / / O ⋆ exp ( G ) _ ⋆   _ ⋆ O O O ( G )  ♥ (5.4) o o O exp ( G ) (5.14) (the numb ers u nder the horizontal arr ows ar e r efer enc es to the formulas in our text ). Example 5.4 . F or the gr oup GL n ( C ) the reﬂexivity diagra m (5.14) is as follows: O ⋆ ( GL n ( C ))  ♥ (4.28) / / R ⋆ ( GL n ( C )) _ ⋆   _ ⋆ O O O ( GL n ( C ))  ♥ (5.2) o o R ( GL n ( C )) (5.15) 96 § 6 Holomorphic reﬂexivit y as a generalization of P on try agin dualit y (a) P on tryagi n dualit y for compactly generated Stein groups The complex circle C × we were ta lking a bout in § 3(a), o ccupies a mong a ll Ab elian c ompactly g enerated Stein groups the same place, as the usual “real” circle T = R / Z among all locally compact Abelian groups (or among a ll Abelian compactly g enerated real Lie g roups), since for the Ab elian compactly g enerated Stein gr oups the following v ariant of Pon tryagin’s duality theory holds. Let G b e an Ab elian compac tly genera ted Stein gr oup. Let us call an ar bitrary holomo rphic homo- morphism of a Stein gr oup G into the co mplex cir cle χ ∈ G • ⇐ ⇒ χ : G → C × a holomorphic char acter on G . The set G • of all holomor phic c haracter s on G is a top ological gro up with resp ect to the p oint wise multiplication a nd the top ology of unifor m conv ergence on compact sets. The following theore m shows that the op eration G 7→ G • is analo gous to the Pontry agin op eration of pas sage to the dual lo cally compact Ab elian group: Theorem 6.1. If G is an Ab elian c omp actly gener ate d St ein gr oup, then its dual gr oup G • is again an Ab elian c omp actly gener ate d gr oup and the mapping i G : G → G •• , i G ( x )( χ ) = χ ( x ) , x ∈ G, χ ∈ G • is an isomorphism of (t op olo gic al gr oups and of ) functors G 7→ G and G 7→ G •• : G •• ∼ = G In view of this fact we call G • dual c omplex gr oup for the g roup G . Pr o of. First we need to note that this is true for the s pecial case s where G = C , C × , Z and for the ca se of a ﬁnite Ab elian group G = F . This is a c orollary o f the following o b vious formulas: C • ∼ = C , ( C × ) • ∼ = Z , Z • ∼ = C × , F • ∼ = F After that it rema ins to note that every compactly genera ted Stein gr oup has the form G ∼ = C l × ( C × ) m × Z n × F ( l , m, n ∈ Z + ) so its dual g roup has the form G • ∼ = C l × Z m × ( C × ) n × F ( l , m, n ∈ Z + ) So G • is an Abe lian compactly generated Stein gr oup. The second dua l g roup G •• turns out to b e isomorphic to G : G •• ∼ = C l × ( C × ) m × Z n × F ∼ = G (b) F ourier transform as an Arens-Mic hael en v elop e If G is an Abelia n compactly gener ated Stein group, then every its holomor phic character χ : G → C × is a holomo rphic function on G . In other words we can think of the dual complex gr oup G • as a subgroup in the gro up of inv ertible element s o f the alge bra O ( G ) of holomorphic functions o n G : G • ⊂ O ( G ) § 6. HOLOMO RPHIC REFLE XIVITY AS A GENERALIZA TION OF PO NTR Y A GIN DUALITY 97 If we pass to dual ob jects by Theorem 6.1, we obta in that the group G itself is included by the trans- formation i G int o the group of invertible elemen ts of the a lgebra O ( G • ) of holo morphic functions on G • : i G : G → G •• ⊂ O ( G • ) . On the other hand, obviously , G is included (through delta-functionals) into algebra O ⋆ ( G ): δ : G → O ⋆ ( G ) ( x 7→ δ x ) . By [1, Theore m 10.1 2] this implies that there e xists a unique homomorphism of stereo t yp e algebra s ♯ G : O ⋆ ( G ) → O ( G • ) , such that the fo llo wing dia gram is commutativ e: G O ⋆ ( G ) O ( G • )   ? ? ? ? ? ? ? i G          δ / / _ _ _ _ _ ♯ G (this is the prope rt y o f b eing group algebra for O ⋆ ( G )). It is natural to call the homomor phism ♯ G : O ⋆ ( G ) → O ( G • ) the (in verse) F ourier tr ansform on the Stein group G , since it is deﬁned by the same formula as for the (inv erse) F ourier transform for meas ures and distr ibutions [16, 3 1.2]: v alue of the func tion α ♯ ∈ C G • in the p oint χ ∈ G • ↓ z }| { α ♯ ( χ ) = α ( χ ) | {z } ↑ action of the fu nctional α ∈ O ⋆ ( G ) on th e fun ction χ ∈ G • ⊆ O ( G ) , χ ∈ G • ( α ∈ O ⋆ ( G ) , w ∈ O ( G • )) (6.1) Theorem 6.2. F or any Ab elian c omp actly gener ate d Stein gr oup G its F ourier tr ansform ♯ G : O ⋆ ( G ) → O ( G • ) , is: (a) a homomorphism of rigid Hopf-F r´ echet algebr as, and (b) an Ar ens- Mich ael envelop e of the algebr a O ⋆ ( G ) . As a c or ol lary the fol lowing isomorphisms of rigid H opf-F r´ echet algebr as hold: O ⋆ exp ( G ) ∼ =  O ⋆ ( G )  ♥ ∼ = O ( G • ) (6.2) and the r eﬂexivity diagr am for G takes the form O ⋆ ( G )  F ourier tr ansform (6.1) / / O ( G • ) _ ⋆   _ ⋆ O O O ( G )  F ourier tr ansform (6.1) o o O ⋆ ( G • ) (6.3) Like Theo rem 6.1, this is prov ed by successive c onsideration o f the cases G = C , C × , Z and the ca se of ar bitrary ﬁnite Ab elian group G = F . In the rest of this s ection up to the “inclusio n diag ram” we devote to this. 98 Finite Ab elian group. As we told in § 3(a), every ﬁnite gro up G can b e considere d a s a complex Lie group (of zer o dimension), on which every function is holomorphic. Moreover, in exa mple § 4(c) we hav e noticed that every function o n G has exp onent ial type, so the algebra s O exp ( G ), O ( G ) and C G coincide: O exp ( G ) = O ( G ) = C G If in addition G is co mm utative, then the theorem 6.2 w e illustrate here turns into a formally more strong prop osition: Prop osition 6.1. If G is a ﬁnite Ab elian gr oup, then the formula (6.1) est ablishes an isomorphism of Hopf algebr as: O ⋆ exp ( G ) = O ⋆ ( G ) = C G ∼ = C G • = O ( G • ) = O exp ( G • ) (6.4) Complex plane C . Let for every λ ∈ C the symbol χ λ denote a character o n the group C , deﬁned by formula: χ λ ( t ) = e λ · t The mapping λ ∈ C 7→ χ λ ∈ C • is an isomor phism of complex gro ups C ∼ = C • and this isomo rphism turns for m ula (6.1 ) into for m ula α ♯ ( λ ) = α ( χ λ ) , λ ∈ C ( α ∈ O ⋆ exp ( C ) , w ∈ O ( C )) (6.5) (w e denote this isomo rphism by the sa me symbol ♯ , a lthough formally it is a co mposition of mapping s (6.1) and λ 7→ χ λ ). As a result Theor em 6.2 b eing applied to the gr oup G = C is turned into Prop osition 6.2. F ormula (6 .5) deﬁnes a homomorphism of st er e otyp e H opf algebr as ♯ C : O ⋆ ( C ) → O ( C ) which is an Ar ens-Michael envelop e of t he algebr a O ⋆ ( C ) , and est ablish es an isomorphism of Hopf-F r´ echet algebr as: O ⋆ exp ( C ) ∼ = O ( C ) (6.6) W e shall need the following Lemma 6.1. Seminorms of the form || α || C = X k ∈ N | α k | · C k , C > 0 , (6.7) (i.e. s p e cial c ase of seminorms (3.73) , when r k = C k k ! ) form a fundamental system in the set of al l submultiplic ative c ontinuous seminorms on O ⋆ ( C ) . Pr o of. As we ha d noted in § 3(c), the m ultiplication in O ( C ) and in O ⋆ ( C ) is deﬁned b y the same for m ulas on s eries (3.7 1). So we c an s a y that the submultiplicativit y o f seminor ms (6 .7) is already proven, since in Ex ample 5.3 we ha d proven the same fact for seminor ms (3.72), deﬁned by the same formula o n the series. Let us show that semino rms (6.7) form a fundamental system among all submu ltiplicative co n tinu ous seminorms on O ⋆ ( C ). T his is done like in Prop osition 3.14. Let p b e a subm ultiplicative con tinuous seminorm: p ( α ∗ β ) 6 p ( α ) · p ( β ) Put r k = 1 k ! p ( ζ k ) § 6. HOLOMO RPHIC REFLE XIVITY AS A GENERALIZA TION OF PO NTR Y A GIN DUALITY 99 Then ( k + l )! · r k + l = p ( ζ k + l ) = p ( ζ k ∗ ζ l ) 6 p ( ζ k ) · p ( ζ l ) = ( k ! · r k ) · ( l ! · r l ) The seq uence A k = r k · k ! sa tisﬁes the r ecurrent ine qualit y A k +1 6 A k · A 1 , which implies A k 6 C k , for C = A 1 . T his in its tur n implies inequalities r k 6 C k k ! Now using the sa me rea sonings as in the pro of of Prop osition 3.14, we obtain: p ( α ) 6 (3.7 6) 6 ||| α ||| r = X k ∈ N r k · | α k | · k ! 6 X k ∈ N | α k | · C k = || α || C Pr o of of Pr op osition 6.2 . Note fro m the very b eginning tha t the mapping ♯ C : O ⋆ ( C ) → O ( C ), deﬁned by F ormula (6.5 ) is co n tinuous: b y the contin uit y o f the mapping λ ∈ C 7→ χ λ ∈ O ( C ), e v ery compact set T in C is turned into a compact set { χ λ ; λ ∈ T } in O ( C ), so if the net of functionals α i tends to zero in O ⋆ ( C ), then for any compac t set T in C we hav e α ♯ i ( λ ) = α i ( χ λ ) ⇒ λ ∈ T 0 , i → ∞ Thu s, the functions α ♯ i tend to zero in O ( C ). F ur ther, note that the mapping ♯ C turns the functionals ζ n int o functions z n : ( ζ n ) ♯ ( λ ) = ζ n ( χ λ ) =  d n d t n e λt       t =0 = λ n = z n ( λ ) F r om this a nd from the contin uit y of ♯ C it follows that the mapping act on functionals α as the substitution of monomials ζ n in the decomp osition (3.70) by monomia ls z n : α ♯ = ∞ X n =0 α n · ζ n ! ♯ = ∞ X n =0 α n · ( ζ n ) ♯ = ∞ X n =0 α n · z n This immediately implies the rest. 1. First, the mapping ♯ C : O ⋆ ( C ) → O ( C ) is an Arens-Michael env elop e, sinc e by Lemma 6.1, every submu ltiplicative contin uous semino rm on O ⋆ ( C ) is ma jorized by a semino rm of the form (6.7), which in its turn can b e extended by the mapping ♯ C to a seminorm (3.72) on O ( C ). 2. Second, the mapping ♯ C : O ⋆ ( C ) → O ( C ) is a homomor phism of algebra s, since by formulas (3.69)- (3.70) these a lgebras can b e considered as algebr as of p ow er ser ies, wher e the multiplication is deﬁned by usual formulas for p o wer series (3.71), and ♯ C will b e just inclusio n of one algebr a into ano ther, mor e wide, algebr a. 3. T o prove that the mapping ♯ C : O ⋆ ( C ) → O ( C ) is an isomorphism of coalg ebras, let us note that the dual mapping ( ♯ C ) ⋆ : O ⋆ ( C ) → ( O ⋆ ( C )) ⋆ = O ( C ) ⋆⋆ coincides with ♯ C up to the isomor phism i O ( C ) : O ( C ) ∼ = O ( C ) ⋆⋆ : O ⋆ ( C ) O ( C ) ⋆⋆ O ( C ) / / ( ♯ C ) ⋆ $ $ J J J J J J ♯ C : : t t t t t t i O ( C ) (6.8) 100 This follows fro m the formula ♯ C ( δ x ) = χ x (6.9) Indeed, ♯ C ( δ x )( λ ) = δ x ( χ λ ) = χ λ ( x ) = e λx = χ x ( λ ) Now we obtain: ♯ ⋆ ( δ a )( δ b ) = δ a ( ♯ C ( δ b )) = (6.9) = δ a ( χ b ) = ♯ C ( δ a )( b ) = δ b ( ♯ C ( δ a )) == i O ( C ) ( ♯ C ( δ a ))( δ b ) = ( i O ( C ) ◦ ♯ )( δ a )( δ b ) This is true for a ll a, b ∈ C . On the other hand, the linear hull o f delta-functionals is dense in O ⋆ , s o ( ♯ C ) ⋆ = i O ( C ) ◦ ♯ C i.e. the diagram (6.8) is commutativ e. But we hav e already pro ved that ♯ C is a homomorphism of algebras , and for i O ( C ) this is obvious. Hence, ( ♯ C ) ⋆ is also a homomor phism of algebras , and this means that ♯ C is a homomor phism of coalg ebras. 4. Now it remains to prove that ♯ C preserves antipo de: σ O ( C ) ( χ λ )( x ) = χ λ ( − x ) = e − λx = ( e λx ) − 1 = χ λ ( x ) − 1 ⇓ σ O ( C ) ( χ λ ) = χ − 1 λ ⇓ ( σ O ⋆ ( C ) ( α )) ♯ ( λ ) = ( σ O ⋆ ( C ) ( α ))( χ λ ) = ( α ◦ σ O ( C ) )( χ λ ) = α ( σ O ( C ) ( χ λ )) = α ( χ − 1 λ ) = α ♯ ( − λ ) = σ O ( C ) ( α ♯ )( λ ) ⇓ ( σ O ⋆ ( C ) ( α )) ♯ = σ O ( C ) ( α ♯ ) Complex circle C × . Let for a n y n ∈ Z the symbol z n denote the character on the gr oup C × , deﬁned by formula z n ( t ) = t n The mapping n ∈ Z 7→ z n ∈ ( C × ) • is a homomor phism of complex gro ups Z ∼ = ( C × ) • and formula (6.1) under this isomorphism takes the form α ♯ ( n ) = α ( z n ) , n ∈ Z ( α ∈ O ⋆ exp ( C × )) (6.10) (like in the pr evious e xample we denote this mapping by the sa me symbol ♯ , although formally this is a comp osition of mappings (6.1) and n 7→ z n ). As a res ult Theorem 6.2, being applied to group G = C × is turned into Prop osition 6.3. F ormula (6 .10) deﬁnes a homomorphism of ster e otyp e Hopf algebr as ♯ C × : O ⋆ ( C × ) → O ( Z ) which is an Ar ens-Michael envelop e of the algebr a O ⋆ ( C × ) , and establishes an isomorphism of Hopf- F r´ echet algebr as O ⋆ exp ( C × ) ∼ = O ( Z ) = C Z (6.11) W e shall need § 6. HOLOMO RPHIC REFLE XIVITY AS A GENERALIZA TION OF PO NTR Y A GIN DUALITY 101 Lemma 6.2. The seminorms of the form || α || N = X | n | 6 N | α n | , N ∈ N (6.12) – i.e. the sp e cial c ase of seminorms (3.48) , when r n = ( 1 , | n | 6 N 0 , | n | > N – form a fundamental system in t he set of al l submult iplic ative c ontinuous seminorms on O ⋆ ( C × ) . Pr o of. Subm ultiplicativity of seminor ms (6.12) follows from the formula for the o pera tion of multiplication in O ⋆ ( C × ): || α ∗ β || N = X | n | 6 N | ( α ∗ β ) n | = (3.46) = X | n | 6 N | α n · β n | 6   X | n | 6 N | α n |   ·   X | n | 6 N | β n |   = || α || N · || β || N Let us s ho w that seminor ms (3 .48) form a fundamental system among all submultiplicativ e contin uous seminorms on O ⋆ ( C × ). Le t p b e a s ubm ultiplicative contin uous seminorm: p ( α ∗ β ) 6 p ( α ) · p ( β ) Put r n = p ( ζ n ) Then r n = p ( ζ n ) = p ( ζ n ∗ ζ n ) 6 p ( ζ n ) · p ( ζ n ) = r 2 n i.e. 0 6 r n 6 r 2 n , hence r n > 1, or r n = 0. B ut by Lemma 3.2 , the num b ers r n m ust s atisfy the condition (3.49), which in its turn implies that r n → 0. This is p ossible only if all those num b ers, ex cept mayb e a ﬁnite subfamily , v a nishes: ∃ N ∈ N ∀ n ∈ Z | n | > N = ⇒ r n = 0 Put M = max n r n , then by Lemma 3 .2 we obta in: p ( α ) 6 ||| α ||| r = X n ∈ Z r n · | α n | = X | n | 6 N r n · | α n | 6 X | n | 6 N M · | α n | = M · || α || N Be ginning of the pr o of of Pr op osition 6.3. Note that the mapping ♯ C × : O ⋆ ( C × ) → O ( Z ), deﬁned by F o rm ula (6 .10) is c on tinuous: if a net of functionals α i tends to zero in O ⋆ ( C × ), then for any n ∈ Z we hav e α ♯ i ( n ) = α i ( z n ) − → 0 , i → ∞ This means that α ♯ i tends to zero in O ( Z ) = C Z . F ur ther, let us note that the mapping ♯ C × turns functiona ls ζ k int o characteristic functions of singletons in Z : ( ζ k ) ♯ ( n ) = ζ k ( z n ) = (3.32) =  1 , n = k 0 , n 6 = k  = (3.16) = 1 k ( n ) F r om this and from the con tinu ity of ♯ C × it follo ws t hat this mapping acts on funct ionals α a s substitution of monomials ζ n in the decomp osition (3.70) by monomia ls 1 n : α ♯ = X n ∈ Z α n · ζ n ! ♯ = X n ∈ Z α n · ( ζ n ) ♯ = X n ∈ Z α n · 1 n 102 This in its turn imply the most part of Pro positio n 6 .3. 1. First, the mapping ♯ C × : O ⋆ ( C × ) → O ( Z ) is an Arens-Michael env elop e, so by Lemma 6.2, every submu ltiplicative contin uous semino rm on O ⋆ ( C × ) is ma jorized by a seminorm of the form (6.1 2), which in its turn ca n b e extended by the mapping ♯ C to a contin uous seminorm on O ( Z ) = C Z . 2. Seco nd, the mapping ♯ C × : O ⋆ ( C × ) → O ( Z ) is a homomor phism of a lgebras, since by the for m ula for m ultiplication (3.46) in O ⋆ ( C × ), this mapping can be represented as a space of t w o-s ided seq uences α n with the co ordinate-w ise multip lication, which is included in the wider spa ce O ( Z ) = C Z of all t wo-sided sequences with the co ordina te-wise multiplication by the mapping ♯ C × . 3. we have to postp one to the next example (at page 1 04) the proo f of the fact that the mapping ♯ C × : O ⋆ ( C × ) → O ( Z ) is an isomo rphism o f coalg ebras. 4. Let us chec k that ♯ C × preserves antipo de: σ O ( C × ) ( z n )( x ) = z n ( x − 1 ) = x − n = z − n ( x ) ⇓ σ O ( C × ) ( z n ) = z − n ⇓ ( σ O ⋆ ( C × ) ( α )) ♯ ( n ) = ( σ O ⋆ ( C × ) ( α ))( z n ) = ( α ◦ σ O ( C × ) )( z n ) = α ( σ C × ( z n )) = α ( z − n ) = α ♯ ( − n ) = σ O ( Z ) ( α ♯ )( n ) ⇓ ( σ O ⋆ ( C × ) ( α )) ♯ = σ O ( Z ) ( α ♯ ) Group of in tegers Z . Let for a n y t ∈ C × the symbol χ t denote a character on the group Z , de ﬁned by formula χ t ( n ) = t n (6.13) The mapping t ∈ C × 7→ χ t ∈ Z • is an isomor phism of complex gro up C × ∼ = Z • and F ormula (6.1) under this iso morphism takes the fo rm α ♯ ( t ) = α ( χ t ) , t ∈ C × ( α ∈ O ⋆ exp ( Z )) (6.14) (again we denote this mapping by the same symbol ♯ , although it is a c ompositio n of the mappings (6.1) and t 7→ χ t ). As a result, Theor em 6.2, b eing applied to gr oup G = Z , is turned into Prop osition 6.4. F ormula (6 .14) deﬁnes a homomorphism of rigid ster e otyp e Hopf algebr as ♯ Z : O ⋆ ( Z ) → O ( C × ) which is an Ar en s - Mich ael envelop e of the algebr a O ⋆ ( Z ) , and establishes an isomorphism of Hopf-F r´ echet algebr as O ⋆ exp ( Z ) ∼ = O ( C × ) (6.15 ) W e need Lemma 6.3. The seminorms of the form || α || C = X n ∈ Z | α n | · C | n | , C > 1 (6.16) (i.e. sp e cial c ase of seminorms (3.24) , when r n = C | n | ) form a fun damental system in the set of al l submultiplic ative c ontinuous seminorms on O ⋆ ( Z ) . § 6. HOLOMO RPHIC REFLE XIVITY AS A GENERALIZA TION OF PO NTR Y A GIN DUALITY 103 Pr o of. W e had a lready noted the submultiplicativit y of seminorms (6.16) in Exa mple 5.2. Let us show that they form a fundamental s ystem. Let p b e a submultiplicativ e contin uous seminorm on on O ⋆ ( Z ): p ( α ∗ β ) 6 p ( α ) · p ( β ) Put r n = p ( δ n ). Then r k + l = p ( δ k + l ) = p ( δ k ∗ δ l ) 6 p ( δ k ) · p ( δ l ) = r k · r l F r om this recurrent formula it follows that r n 6 M · C | n | where M = r 0 , C = max { r 1 ; r − 1 } , and now by Lemma 3.1 we obtain: p ( α ) 6 (3.28) 6 ||| α ||| r = X n ∈ Z r n · | α n | 6 X n ∈ Z M · C | n | · | α n | = M · || α || C Pr o of of Pr op osition 6.4 . First of all, let us note that the mapping ♯ Z : O ⋆ ( Z ) → O ( C × ), deﬁned b y formula (6 .14) is contin uous: by c on tinuit y of the mapping t ∈ C × 7→ χ t ∈ O ( Z ), every compact se t T in C × is turned into a compact set { χ t ; t ∈ T } in O ( Z ), so if a net of functiona ls α i turns to zero in O ⋆ ( Z ), then for any compact set T in C × we hav e α ♯ i ( t ) = α i ( χ t ) ⇒ λ ∈ T 0 , i → ∞ This means that α ♯ i tends to zero in O ( C × ). F ur ther, let us note that the mapping ♯ Z turns functionals δ n int o mo nomials z n : ( δ n ) ♯ ( t ) = δ n ( χ t ) = χ t ( n ) = t n = z n ( t ) This together with contin uit y of ♯ Z implies that this mapping acts on functionals α as substitution of monomials δ n in the decomp osition (3.18) by monomia ls z n : α ♯ = X n ∈ Z α n · δ n ! ♯ = X n ∈ Z α n · ( δ n ) ♯ = X n ∈ Z α n · z n This implies the r est. 1. First, the ma pping ♯ Z : O ⋆ ( Z ) → O ( C × ) is an Arens-Michael env elope , so by Le mma 6.3, every submu ltiplicative seminor m on O ⋆ ( Z ) is ma jorized by a semino rm o f the for m (6.1 6), which in its turn can b e extended by the mapping ♯ Z to a seminorm (3.47) on O ( C × ). 2. Second, the mapping ♯ Z : O ⋆ ( Z ) → O ( C × ) is a homomor phism of algebra s, since by (3.19)-(3.46) these are alg ebras of p ow er series with the usua l mu ltiplication of p ow er s eries, and ♯ C is simply inclusion of one algebr a into another, wider alg ebra. 3. T o pr o ve that the ma pping ♯ Z : O ⋆ ( Z ) → O ( C × ) is an isomor phism of coalg ebras, let us note that the dual mapping ( ♯ Z ) ⋆ : O ⋆ ( C × ) → ( O ⋆ ( Z )) ⋆ = O ( Z ) ⋆⋆ up to the iso morphism i O ( Z ) : O ( Z ) ∼ = O ( Z ) ⋆⋆ coincides with ♯ C × : O ⋆ ( C × ) O ( Z ) ⋆⋆ O ( Z ) / / ( ♯ Z ) ⋆ $ $ J J J J J J ♯ C × : : t t t t t t i O ( Z ) (6.17) 104 This follows fro m formula δ t ( ♯ Z ( δ n )) = t n = δ n ( ♯ C × ( δ t )) , t ∈ C × , n ∈ Z ( 6.1 8) Indeed, δ t ( ♯ Z ( δ n )) = ♯ Z ( δ n )( t ) = δ n ( χ t ) = χ t ( n ) = t n and δ n ( ♯ C × ( δ t )) = ♯ C × ( δ t )( n ) = δ t ( z n ) = z n ( t ) = t n Now we obtain: for t ∈ C × and n ∈ Z ( ♯ Z ) ⋆ ( δ t )( δ n ) = δ t ( ♯ Z ( δ n )) = (6.18) = δ n ( ♯ C × ( δ t )) = i O ( Z ) ( ♯ C × ( δ t ))( δ n ) = ( i O ( Z ) ◦ ♯ C × )( δ t )( δ n ) This is true for any t ∈ C × and for a n y n ∈ Z . On the other hand, delta -functionals gene rate a dense subspace in O ⋆ , hence ( ♯ Z ) ⋆ = i O ( Z ) ◦ ♯ C × i.e. diagram (6 .17) is commutativ e. Now we can note that in the ﬁr st par t of pr oo f of P ropo sition 6.3 (p.102) we hav e alr eady veriﬁed that ♯ C × is a homomor phism o f alg ebras. Certainly , the mapping i O ( Z ) is also a ho momorphism, so we conclude tha t ( ♯ Z ) ⋆ is a homomo rphism of algebra s, and this means that ♯ Z is a homomor phism of coalg ebras. 4. Now it remains to chec k tha t ♯ Z preserves antipo de: σ O ( Z ) ( χ t )( n ) = χ t ( − n ) = t − n = ( t − 1 ) n = χ t − 1 ( n ) ⇓ σ O ( Z ) ( χ t ) = χ t − 1 ⇓ ( σ O ⋆ ( Z ) ( α )) ♯ ( t ) = ( σ O ⋆ ( Z ) ( α ))( χ t ) = ( α ◦ σ O ( Z ) )( χ t ) = α ( σ O ( Z ) ( χ t )) = α ( χ t − 1 ) = α ♯ ( t − 1 ) = σ O ( C × ) ( α ♯ )( t ) ⇓ ( σ O ⋆ ( Z ) ( α )) ♯ = σ O ( C × ) ( α ♯ ) End of the pr o of of Pr op osition 6.3. In Prop osition 6.3 it remains to pro of that the ma pping ♯ C × : O ⋆ ( C × ) → O ( Z ) is an isomo rphism o f coalge bras. Note that the dual mapping ( ♯ C × ) ⋆ : O ⋆ ( Z ) → ( O ⋆ ( C × )) ⋆ = O ( C × ) ⋆⋆ coincides with ♯ Z up to the iso morphism i O ( C × ) : O ( C × ) ∼ = O ( C × ) ⋆⋆ : O ⋆ ( Z ) O ( C × ) ⋆⋆ O ( C × ) / / ( ♯ C × ) ⋆ $ $ J J J J J J J ♯ Z : : t t t t t t t i O ( C × ) (6.19) This also follows fro m (6.18): if t ∈ C × and n ∈ Z ( ♯ C × ) ⋆ ( δ n )( δ t ) = δ n ( ♯ C × ( δ t )) = (6.18) = δ t ( ♯ Z ( δ n )) = i O ( C × ) ( ♯ Z ( δ n ))( δ t ) = ( i O ( Z ) ◦ ♯ C × )( δ n )( δ t ) This is true for a ll t ∈ C × and n ∈ Z , and the delta-functionals form a dense subspace in O ⋆ , so ( ♯ C × ) ⋆ = i O ( C × ) ◦ ♯ Z i.e. Diagram (6.19) is commutativ e. Now we can note that in pro of o f Pr opo sition 6.4 we hav e alr eady veriﬁed that ♯ Z is a homomor phism of algebras. And for i O ( C × ) this is obvious, so we obtain that ( ♯ C × ) ⋆ is also a homo morphism of alg ebras, and this mea ns that ♯ C × is a homomor phism of coalg ebras. § 6. HOLOMO RPHIC REFLE XIVITY AS A GENERALIZA TION OF PO NTR Y A GIN DUALITY 105 Pro of of Theorem 6.2 Now formulas (6.4), (6.6), (6.11), (6.1 5) pro ve isomorphism (6.2) for that cases G = C , C × , Z and for the case of ﬁnite gro up G , so it remains just to apply for m ulas (4.4 3) and (4.44): let us decomp ose an Ab elian compactly genera ted Stein gr oup G into a dir ect pro duct G = C l × ( C × ) m × Z n × F where F is a ﬁnite group. Then we hav e: O ⋆ exp ( G ) = O ⋆ exp ( C l × ( C × ) m × Z n × F ) = (4.43) = = O ⋆ exp ( C ) ⊙ l ⊙ O ⋆ exp ( C × ) ⊙ m ⊙ O ⋆ exp ( Z ) ⊙ n ⊙ O ⋆ exp ( F ) = = O ( C • ) ⊙ l ⊙ O (( C × ) • ) ⊙ m ⊙ O ( Z • ) ⊙ n ⊙ O ( F • ) = = (4.43) = O (( C • ) l × (( C × ) • ) m × ( Z • ) n × F • ) = O ( G • ) (c) Inclusion diagram Theorem 6.3. t he fol lowing c onst ruction is a gener alization of t he Pontryagin duality the ory fr om the c ate gory of Ab elian c omp actly gener ate d S tein gr oups to the c ate gory of c omp actly gener ate d Stein gr oups with the algebr aic c onne cte d c omp onent of identity, holomorphically reﬂexive Hopf algebras H 7→ ( H ♥ ) ⋆ / / holomorphically reﬂexive Hopf algebras compactly generated Stein groups with algebraic co m p onent of identi ty O ⋆ ( G ) 7→ G O O compactly generated Stein group s with alg e braic comp onent of identi ty O ⋆ ( G ) 7→ G O O Abel ian compactly generated Stein groups O O G 7→ G • / / Abel ian compactly generated Stein group s O O and t he c ommutativity of this diagr am is establishe d by t he isomorphism of functors O ⋆ ( G • ) ∼ =   O ⋆ ( G )  ♥  ⋆ . Pr o of. In Theorem 5.4 we have alrea dy show ed that the functor G 7→ O ⋆ ( G ) a cts fro m the catego ry of compactly generated Stein groups with alge braic comp onent of identit y into the catego ry of holomo r- phically reﬂexive rig id stereotype Hopf algebras. So we need o nly to verify the co mm utativity o f this “categor ical diagra m”. This follows from Theorem 6.1: if G is an Abelia n compactly ge nerated Stein group, then the walk around the diagram gives the following o b jects: O ⋆ ( G )  ♥ / / O ⋆ exp ( G ) ∼ = (6.2) ∼ = O ( G • )  ⋆ / / O ⋆ ( G • ) G _ O ⋆ O O  • / / G • _ O ⋆ O O 106 § 7 App end ix: holomorphic reﬂexivit y of the quan tum group ‘ az + b ’ In this ﬁnal section we, following o ur promises in Introduction, will show by the e xample o f the quantum group ‘ az + b ’, that the holomor phic reﬂexivity des cribed ab ov e do es not restrict itself on algebras of analytical functionals O ⋆ ( G ), but lengthens into the theory o f q uan tum g roups. (a) Quan tum com binatorial form ulas The theor y of qua n tum gr oups has its own analo g of elemen tary co m binatorics, used in the situations where the computations ar e applied to v ariables with the following commutation law y x = q xy (7.1) where q is a ﬁxed num be r. F or those computations in par ticular, the analo gs o f usual binomial formulas are deduced. Some of them will b e useful for us in our constructions c onnected with ‘ az + b ’, s o w e reco rd them for further refer ences (we r efer the r eader to deta ils in C. Ka ssel’s textb oo k [19]). F o r a n arbitr ary p ositive integer n we put ( n ) q := 1 + q + ... + q n − 1 = q n − 1 q − 1 , ( n )! q := (1) q (2) q ... ( n ) q = ( q − 1)( q 2 − 1) ... ( q n − 1) ( q − 1) n (7.2) The integer ( n )! q will b e called quantum factorial o f the integer n . The quantum binomial c o eﬃcient is deﬁned by for m ula  n k  q := ( ( n )! q ( k )! q · ( n − k )! q , k 6 n 0 , k > n (7.3) Theorem 7.1 ( quan tum binomi al form ula). L et x and y b e elements of an asso ciative algebr a A satisfying c ondition (7.1) . Then for al l n ∈ N ( x + y ) n = n X k =0  n k  q · x k · y n − k (7.4) Theorem 7.2 ( quan tum Chu-W andermond form ula). F or al l l , m, n ∈ N  m + n l  q = X max { 0 ,l − n } 6 i 6 min { l,m } q ( m − i ) · ( l − i ) ·  m i  q ·  n l − i  q = = X 0 6 i 6 l q ( m − i ) · ( l − i ) ·  m i  q ·  n l − i  q (7.5) Pr o of. 6 If l > m + n , then for any i = 0 , ..., m we hav e n < l − m 6 l − i , s o  n l − i  q = 0. Thus, b oth sums in (7.5) v anish. And the same happ ens with  m + n l  q , so formula (7.5) is trivia l. Thu s only the case of l 6 m + n is interesting. Co nsider the equality ( x + y ) m + n = ( x + y ) m ( x + y ) n . Removing the par en thesis in (7 .4) we obtain m + n X l =0  m + n l  q · x l · y n − l = m X i =0  m i  q · x i · y m − i ! ·   n X j =0  n j  q · x j · y n − j   = 6 W e give here the pro of of the Chu-W andermond formula just to draw reader’s atten tion to the limits of summing max { 0 , l − n } 6 i 6 min { l, m } , which will b e useful below. § 7. APPE NDIX: HOLO MORPHIC REFLEXIVITY OF THE QUANTUM GR OUP ‘ AZ + B ’ 107 = m X i =0 n X j =0  m i  q ·  n j  q · x i · y m − i · x j · y n − j = (7.1) = m X i =0 n X j =0  m i  q ·  n j  q · q ( m − i ) · j · x i + j · y m + n − i − j Let us consider in the last s um the terms with indices i and j connected by equality i + j = l . All those terms can be indexed by the para meter i , if we express j trough i by for m ula j = l − i . W e need o nly to note that to obtain a bijection the index i must v ary in the following limits: max { 0 , l − n } 6 i 6 min { l, m } This follows fro m the res trictions on i a nd j : ( 0 6 i 6 m 0 6 j 6 n ⇐ ⇒ ( 0 6 i 6 m 0 6 l − i 6 n ⇐ ⇒ ( 0 6 i 6 m − n 6 i − l 6 0 ⇐ ⇒ ( 0 6 i 6 m l − n 6 i 6 l Now equating the co eﬃcients at monomia l x l · y n − l , we obtain the ﬁrst e qualit y in (7.5):  m + n l  q = X max { 0 ,l − n } 6 i 6 min { l,m } q ( m − i ) · ( l − i ) ·  m i  q ·  n l − i  q The second eq ualit y is e viden t, since for i < l − n and i > l we hav e res pectively n < l − i or l − i < 0 , hence  n l − i  q = 0, and the ter ms v anish: X max { 0 ,l − n } 6 i 6 min { l,m } = X 0 6 i< max { 0 ,l − n } | {z } 0 + X max { 0 ,l − n } 6 i 6 min { l,m } + X min { l,m } 0 (7.17) antip o de: σ ( a ⊙ t k ) = ( − 1) k · q − k ( k +1) 2 · z − k · ( M ⋆ ω ) k ( σ H ( a )) ⊙ t k (7.18) H ⊙ R ( C ) with such a structu r e of Hopf algebr a is denote d by H z ⊙ ω R ( C ) ; the c ommon formula for multiplic ation in this algebr a has the form: u · v = X k ∈ N u k ⊙ t k ! · X l ∈ N v l ⊙ t l ! = X m ∈ N m X k =0 u k · ( M ⋆ ω ) k ( v m − k ) ! ⊙ t m (7.19) 110 (b) the tensor pr o duct H ⋆ ⊛ R ⋆ ( C ) has a u nique struct u r e of pr oje ctive Hopf algebr a with t he algebr aic op er ations deﬁne d by formulas multiplic ation: α ⊛ τ k ∗ β ⊛ τ l = α · ( M ⋆ z ) k ( β ) ⊛ τ k + l (7.20) unit: 1 H ⋆ ⊛ R ( C ) = 1 H ⋆ ⊛ 1 R ⋆ ( C ) (7.21) c omu ltipli c ation: κ ( α ⊛ τ k ) = k X i =0  k i  q · θ   id H ⋆ ⊛ M i ω  ( κ H ⋆ ( α )) ⊛ τ i ⊛ τ k − i  = (7.22) = X ( α ) k X i =0  k i  q · α ′ ⊛ τ i ⊛ ( ω i ∗ α ′′ ) ⊛ τ k − i c oun it: ε ( α ⊛ τ k ) = ( ε H ⋆ ( α ) , k = 0 0 , k > 0 (7.23) antip o de: σ ( α ⊛ τ k ) = ( − 1) k · q − k ( k +1) 2 · ω − k ∗ ( M ⋆ z ) k ( σ H ⋆ ( α )) ⊛ τ k (7.24) H ⋆ ⊛ R ⋆ ( C ) with such a structu r e of H opf algebr a is denote d by H ⋆ ω ⊛ z R ⋆ ( C ) ; the c ommon formula for multiplic ation in this algebr a has t he form: α ∗ β = X k ∈ N α k ⊙ τ k ! · X l ∈ N β l ⊙ τ l ! = X m ∈ N m X k =0 α k ∗ ( M ⋆ z ) k ( β m − k ) ! ⊙ τ m (7.25) (c) the biline ar form * X k ∈ N u k ⊙ t k , X k ∈ N α k ⊛ τ k + = X k ∈ N h u k , α k i · ( k )! q (7.26) turns H z ⊙ ω R ( C ) and H ⋆ ω ⊛ z R ⋆ ( C ) into a dual p air of ster e otyp e Hopf algebr as:  H z ⊙ ω R ( C )  ⋆ ∼ = H ⋆ ω ⊛ z R ⋆ ( C ) (7.27) W e divide the pr oo f of this theo rem in 7 lemmas. Some of them a re evident, and in those cases we omit the pro of. Lemma 7.1. The multiplic ation and t he un it (7.14) , (7.15) endow H ⊙ R ( C ) with the struct u r e of inje ctive ster e otyp e algebr a, isomorphic to the algebr a of skew p olynomials with c o eﬃcients in algebr a H and the gener ating automorphism ϕ = M ⋆ ω Lemma 7.2. The m ultiplic ation and the u n it (7.20) , (7.21) endow H ⋆ ⊛ R ⋆ ( C ) with the stru ctur e of pr oje ctive ster e otyp e algebr a, isomorphi c t o the algebr a of skew p ower series with c o eﬃcients in algebr a H ⋆ and t he gener ating automorphism ϕ = M ⋆ z Lemma 7.3. The biline ar form (7.26) t urns the c omultiplic ation (7.16) into the multiplic ation (7.20) , and the c ounit (7.17) into t he c ounit (7.21) : h κ ( u ) , α ⊛ β i = h u, α ∗ β i , ε ( u ) = h u, 1 H ⋆ ⊛ 1 R ⋆ ( C ) i (7.28) As a c or ol lary, the c omu lt ipli c ation (7.16) and the c oun it (7.1 7) deﬁne the stru ctur e of inje ctive s t er e otyp e c o algebr a on H ⊙ R ( C ) . § 7. APPE NDIX: HOLO MORPHIC REFLEXIVITY OF THE QUANTUM GR OUP ‘ AZ + B ’ 111 Lemma 7. 4. The biline ar form (7.26) turns multiplic ation (7.14) into c omultiplic ation (7.22) , and un it (7.15) into c ounit (7.23) : h u · v , α i = h u ⊙ v , κ ( α ) i , h 1 H ⊙ 1 R ( C ) , α i = ε ( α ) (7.29) As a c or ol lary, c omultiplic ation (7.22) and c ounit (7.23) deﬁne the structur e of pr oje ctive ster e otyp e c o algebr a on H ⋆ ⊛ R ⋆ ( C ) . Pr o of of L emmas 7.3 and 7.4. Beca use of the symmetry be t ween formulas (7.16)-(7.20) and (7.14)-(7.22) it is suﬃcient to prove (7.28). F or this we can take u = a ⊙ t k . Then the second equality beco mes evident ε ( a ⊙ t k ) =  ε H ( a ) , k = 0 0 , k > 0  = h a ⊙ t k , 1 H ⋆ ⊛ 1 R ⋆ ( C ) i and the ﬁrst one is pr o ved by the following chain:  κ ( a ⊙ t k ) , α ⊛ β  = k X i =0  k i  q · * θ  (1 H ⊙ M i z )( κ H ( a )) ⊙ t i ⊙ t k − i  , X l ∈ N α l ⊛ τ l ⊛ X m ∈ N β m ⊛ τ m + = = X l,m ∈ N k X i =0  k i  q · D θ  (1 H ⊙ M i z )( κ H ( a )) ⊙ t i ⊙ t k − i  , α l ⊛ τ l ⊛ β m ⊛ τ m E = = X l,m ∈ N k X i =0  k i  q · D (1 H ⊙ M i z )( κ H ( a )) ⊙ t i ⊙ t k − i , θ  α l ⊛ τ l ⊛ β m ⊛ τ m E = = X l,m ∈ N k X i =0  k i  q ·  (1 H ⊙ M i z )( κ H ( a )) ⊙ t i ⊙ t k − i , α l ⊛ β m ⊛ τ l ⊛ τ m  = = k X i =0  k i  q · ( i )! q · ( k − i )! q  (1 H ⊙ M i z )( κ H ( a )) , α i ⊛ β k − i  = = ( k )! q · k X i =0  κ H ( a ) , α i ⊛ ( M ⋆ z ) i ( β k − i )  = ( k )! q · k X i =0  a, α i ∗ ( M ⋆ z ) i ( β k − i )  = = * a ⊙ t k , k X i =0 α i ∗ ( M ⋆ z ) i ( β k − i ) ⊛ τ k + = * a ⊙ t k , X n ∈ N n X i =0 α i ∗ ( M ⋆ z ) i ( β n − i ) ! ⊛ τ n + = =  a ⊙ t k , α ∗ β  Lemma 7.5. Comultiplic ation (7.1 6) and c oun it (7.17) ar e homomorphisms of inje ctive ster e otyp e alge- br as, and as a c or ol lary endow H ⊙ R ( C ) with t he stru ctur e of inje ctive ster e otyp e bialgebr a. Lemma 7.6. Comultiplic ation (7.22) and c ounit (7.23) ar e homomorp hisms of pr oje ctive ster e otyp e algebr as, and as a c or ol lary endow H ⋆ ⊛ R ⋆ ( C ) with t he stru ctur e of pr oje ctive st er e otyp e bialgebr a. Pr o of. Again due to the symmetry of formulas it is suﬃcie n t here to prov e the ﬁr st lemma. W e will just chec k that the co m ultiplication (7.16) is a homomorphism of a lgebras (and the reader is suppo sed to chec k the ide n tit y for co unit by a nalogy). Let us note the fo llo wing ide n tities: κ ( a ⊙ 1 · b ⊙ 1) = κ ( a ⊙ 1) · κ ( b ⊙ 1) (7.30) κ (1 ⊙ t k · 1 ⊙ t l ) = κ (1 ⊙ t k ) · κ (1 ⊙ t l ) (7.3 1) κ (1 ⊙ t k · a ⊙ 1) = κ (1 ⊙ t k ) · κ ( a ⊙ 1) (7.32) 112 κ ( a ⊙ 1 · 1 ⊙ t k ) = κ ( a ⊙ 1) · κ (1 ⊙ t k ) (7.33) Indeed, for (7.30) we have: κ ( a ⊙ 1) · κ ( b ⊙ 1 ) = X ( a ) ( a ′ ⊙ 1 ⊙ a ′′ ⊙ 1) · X ( b ) ( b ′ ⊙ 1 ⊙ b ′′ ⊙ 1) = = X ( a ) , ( b ) a ′ b ′ ⊙ 1 ⊙ a ′′ b ′′ ⊙ 1 = κ (( a · b ) ⊙ 1) = κ ( a ⊙ 1 · b ⊙ 1) F o r (7.3 1): κ (1 ⊙ t k ) · κ (1 ⊙ t k ) = k X i =0  k i  q · 1 ⊙ t i ⊙ z i ⊙ t k − i · l X j =0  l j  q · 1 ⊙ t j ⊙ z j ⊙ t l − j = = k X i =0 l X j =0  k i  q ·  l j  q ·  (1 ⊙ t i ) · (1 ⊙ t j )  ⊙  ( z i ⊙ t k − i ) · ( z j ⊙ t l − j )  = (7.14) = = k X i =0 l X j =0  k i  q ·  l j  q ·  1 ⊙ t i + j  ⊙  ( z i · ( M ⋆ ω ) k − i ( z j ) ⊙ t k + l − i − j  = (7.12) = = k X i =0 l X j =0  k i  q ·  l j  q · q ( k − i ) j · 1 ⊙ t i + j ⊙ z i + j ⊙ t k + l − i − j = = i + j = m j = m − i 0 6 m 6 k + l max { 0 , m − l } 6 i 6 min { k, m } ! = = k + l X m =0 X max { 0 ,m − l } 6 i 6 min { k,m }  k i  q ·  l m − i  q · q ( k − i )( m − i ) · 1 ⊙ t m ⊙ z m ⊙ t k + l − m = (7.5) = = k + l X m =0  k + l m  q · 1 ⊙ t m ⊙ z m ⊙ t k + l − m = κ (1 ⊙ t k + l ) = κ (1 ⊙ t k · 1 ⊙ t l ) F o r (7.3 2): κ (1 ⊙ t k ) · κ ( a ⊙ 1 ) = k X i =0  k i  q · 1 ⊙ t i ⊙ z i ⊙ t k − i · X ( a ) a ′ ⊙ 1 ⊙ a ′′ ⊙ 1 = = k X i =0 X ( a )  k i  q ·  (1 ⊙ t i ) · ( a ′ ⊙ 1)  ⊙  ( z i ⊙ t k − i ) · ( a ′′ ⊙ 1)  = = k X i =0 X ( a )  k i  q · ( M ⋆ ω ) i ( a ′ ) ⊙ t i ⊙ z i ( M ⋆ ω ) k − i ( a ′′ ) ⊙ t k − i = (1.40) = = κ (( M ⋆ ω ) k ( a ) ⊙ t k ) = (7.14) = κ (1 ⊙ t k · a ⊙ 1) F o r (7.3 3): κ ( a ⊙ 1) · κ (1 ⊙ t k ) = X ( a ) a ′ ⊙ 1 ⊙ a ′′ ⊙ 1 · k X i =0  k i  q · 1 ⊙ t i ⊙ z i ⊙ t k − i = = X ( a ) k X i =0  k i  q ·  ( a ′ ⊙ 1) · (1 ⊙ t i )  ⊙  ( a ′′ ⊙ 1) · ( z i ⊙ t k − i )  = § 7. APPE NDIX: HOLO MORPHIC REFLEXIVITY OF THE QUANTUM GR OUP ‘ AZ + B ’ 113 = X ( a ) k X i =0  k i  q · a ′ ⊙ t i ⊙ a ′′ z i ⊙ t k − i = (7.16) = κ ( a ⊙ t k ) = κ ( a ⊙ 1 · 1 ⊙ t k ) F r om (7.3 0)-(7.33) it fo llo ws tha t (7.1 6) is a homomorphism a algebra s: κ ( a ⊙ t k · b ⊙ t l ) = κ ( a · ( M ⋆ ω ) k ( b ) ⊙ t k + l ) = κ ( a · ( M ⋆ ω ) k ( b ) ⊙ 1 · 1 ⊙ t k + l ) = (7.33) = = κ ( a · ( M ⋆ ω ) k ( b ) ⊙ 1) · κ (1 ⊙ t k + l ) = κ ( a ⊙ 1 · ( M ⋆ ω ) k ( b ) ⊙ 1) · κ (1 ⊙ t k · 1 ⊙ t l ) = = (7.30) , (7.31) = κ ( a ⊙ 1) · κ (( M ⋆ ω ) k ( b ) ⊙ 1) · κ (1 ⊙ t k ) · κ (1 ⊙ t l ) = = (7.33) = κ ( a ⊙ 1) · κ (( M ⋆ ω ) k ( b ) ⊙ 1 · 1 ⊙ t k ) · κ (1 ⊙ t l ) = = κ ( a ⊙ 1) · κ (1 ⊙ t k · b ⊙ 1) · κ (1 ⊙ t l ) = = (7.32) = κ ( a ⊙ 1) · κ (1 ⊙ t k ) · κ ( b ⊙ 1) · κ (1 ⊙ t l ) = (7.33) = = κ ( a ⊙ 1 · 1 ⊙ t k ) · κ ( b ⊙ 1 · 1 ⊙ t l ) = κ ( a ⊙ t k ) · κ ( b ⊙ t l ) Lemma 7.7. F ormulas (7 .18) and (7.24) deﬁne antip o des in bialgebr as H ⊙ R ( C ) and H ⋆ ⊛ R ⋆ ( C ) , dual to e ach other with r esp e ct to the biline ar form (7.26) : h σ ( u ) , α i = h u, σ ( α ) i (7.34) Pr o of. Let us show that for m ula (7.18) deﬁnes a n antipo de in H ⊙ R ( C ). First we need to verify that σ is an automor phism: σ ( a ⊙ t k · b ⊙ t l ) = σ ( a · ( M ⋆ ω ) k ( b ) ⊙ t k + l ) = = ( − 1) k + l · q − ( k + l )( k + l +1) 2 · z − k − l · ( M ⋆ ω ) k + l σ H  a · ( M ⋆ ω ) k ( b )  ! ⊙ t k + l = = ( − 1) k + l · q − ( k + l )( k + l +1) 2 · z − k − l · ( M ⋆ ω ) k + l σ H  ( M ⋆ ω ) k ( b )  · σ H ( a ) ! ⊙ t k + l = = ( − 1) k + l · q − ( k + l )( k + l +1) 2 · z − k − l · ( M ⋆ ω ) l  ( M ⋆ ω ) k ◦ σ H ◦ ( M ⋆ ω ) k  ( b ) ! · ( M ⋆ ω ) k + l ( σ H ( a )) ⊙ t k + l = = (1.38) = ( − 1) k + l · q − k 2 + l 2 + k + l 2 · q − kl · z − k − l · ( M ⋆ ω ) l ( σ H ( b )) · ( M ⋆ ω ) k + l ( σ H ( a )) ⊙ t k + l = = ( − 1) k + l · q − k ( k +1)+ l ( l +1) 2 · z − l · ( M ⋆ ω ) l ( σ H ( b )) · ( M ⋆ ω ) l  z − k ( M ⋆ ω ) k ( σ H ( a ))  ⊙ t k + l = = ( − 1) l · q − l ( l +1) 2 · z − l · ( M ⋆ ω ) l ( σ H ( b )) ⊙ t l · ( − 1 ) k · q − k ( k +1) 2 · z − k · ( M ⋆ ω ) k ( σ H ( a )) ⊙ t k = = σ ( b ⊙ t l ) · σ ( a ⊙ t k ) 114 Now let us note that diag ram (1.18) b ecomes co mm utative if we put there a ⊙ 1: P ( a ) a ′ ⊙ 1 R ( C ) ⊙ a ′′ ⊙ 1 R ( C ) & σ ⊗ 1 H , , P ( a ) σ H ( a ′ ) ⊙ 1 R ( C ) ⊙ a ′′ ⊙ 1 R ( C )  µ - - P ( a ) σ H ( a ′ ) · a ′′ ⊙ 1 R ( C ) k a ⊙ 1 M κ : : q κ $ $  ε / / ε H ( a )  ι / / ε H ( a ) · 1 H ⊙ 1 R ( C ) k P ( a ) a ′ · σ H ( a ′′ ) ⊙ 1 R ( C ) P ( a ) a ′ ⊙ 1 R ( C ) ⊙ a ′′ ⊙ 1 R ( C )  1 H ⊗ σ 2 2 P ( a ) a ′ ⊙ 1 R ( C ) ⊙ σ H ( a ′′ ) ⊙ 1 R ( C ) 0 µ 1 1 or 1 ⊙ t : 1 ⊙ 1 ⊙ 1 ⊙ t + 1 ⊙ t ⊙ z ⊙ 1 * σ ⊗ 1 H * * 1 ⊙ 1 ⊙ 1 ⊙ t − q − 1 z − 1 ⊙ t ⊙ z ⊙ 1  µ / / 1 ⊙ t − q − 1 q · z − 1 z ⊙ t k 1 ⊙ t F κ 6 6 x κ ( (  ε / / 0  ι / / 0 k − q − 1 z − 1 ⊙ t + q − 1 · z − 1 ⊙ t 1 ⊙ 1 ⊙ 1 ⊙ t + 1 ⊙ t ⊙ z ⊙ 1  1 H ⊗ σ 4 4 − q − 1 1 ⊙ 1 ⊙ z − 1 ⊙ t + 1 ⊙ t ⊙ z − 1 ⊙ 1 , µ 0 0 By Lemma on an tip o de 1.1 this implies that the diagram (1.18) is co mm utative if we put there after v ar ious pr oducts of a ⊙ 1 and 1 ⊙ t . In particular, if we put a ⊙ t k . This means that 1.1 is c omm utative (with arbitr ary argument), and we o btain that the mapping (7.18) is an antipo de in H ⊙ R ( C ). F ur ther, by the symmetry of formulas, the mapping (7.24) is an antipo de in H ⋆ ⊛ R ⋆ ( C ). It remains to verify that the bilinear form (7.26) turns the antipo de (7.18) into the antipo de (7.24). Clearly , formula (7.34) is equiv alent to formula h σ ( a ⊙ t k ) , α ⊛ τ l i = h a ⊙ t k , σ ( α ⊛ τ l ) i F o r k 6 = l b oth sides here v anis h, so we need chec k only the case k = l . Indeed, h σ ( a ⊙ t k ) , α ⊛ τ k i = (7.18) = h ( − 1) k · q − k ( k +1) 2 · z − k · ( M ⋆ ω ) k ( σ H ( a )) ⊙ t k , α ⊛ τ k i = = ( − 1) k · q − k ( k +1) 2 · h z − k · ( M ⋆ ω ) k ( σ H ( a )) , α i · ( k )! q = ( − 1) k · q − k ( k +1) 2 · h ( M ⋆ ω ) k ( σ H ( a )) , ( M ⋆ z − 1 ) k ( α ) i · ( k )! q = = ( − 1) k · q − k ( k +1) 2 · h σ H ( a ) , ω k ∗ ( M ⋆ z − 1 ) k ( α ) i · ( k )! q = ( − 1) k · q − k ( k +1) 2 · h a, σ ⋆ H  ( ω k ∗ ( M ⋆ z − 1 ) k ( α )  i · ( k )! q = § 7. APPE NDIX: HOLO MORPHIC REFLEXIVITY OF THE QUANTUM GR OUP ‘ AZ + B ’ 115 = ( − 1) k · q − k ( k +1) 2 · h a, ω − k ∗ σ ⋆ H  ( M ⋆ z − 1 ) k ( α )  i · ( k )! q = (1.38) = = ( − 1) k · q − k ( k +1) 2 · h a, ω − k ∗ ( M ⋆ z ) k ( σ ⋆ H ( α )) i · ( k )! q = ( − 1) k · q − k ( k +1) 2 · h a ⊙ t k , ω − k ∗ ( M ⋆ z ) k ( σ ⋆ H ( α )) ⊛ τ k i = = (7.24) = h a ⊙ t k , σ ( α ⊛ τ k ) i Chains H ⊙ z ω R ( C ) ⊂ H ⊙ z ω O ( C ) ⊂ H ⊙ z ω O ⋆ ( C ) ⊂ H ⊙ z ω R ⋆ ( C ) and H ⊛ z ω R ( C ) ⊂ H ⊛ z ω O ( C ) ⊂ H ⊛ z ω O ⋆ ( C ) ⊂ H ⊛ z ω R ⋆ ( C ) . The sa me fo rm ulas a nd r easonings as we used in the pr oo f o f Theorem 7 .6, allow us to deﬁne, apar t from H z ⊙ ω R ( C ) and H ⋆ ω ⊛ z R ⋆ ( C ), a serie s of similar ster eot yp e Hopf alge bras. These are algebr as of — skew p olynomials H z ⊙ ω R ( C ) and H z ⊛ ω R ( C ), — skew entire functions H z ⊙ ω O ( C ) and H z ⊛ ω O ( C ), — skew ana lytic functionals H z ⊙ ω O ⋆ ( C ) and H z ⊛ ω O ⋆ ( C ), — skew p ow er ser ies H z ⊙ ω R ⋆ ( C ) and H z ⊛ ω R ⋆ ( C ). Visually the co nnection b et ween them can b e illustrated by the following chains of inclusio ns: H z ⊙ ω R ( C ) ⊂ H z ⊙ ω O ( C ) ⊂ H z ⊙ ω O ⋆ ( C ) ⊂ H z ⊙ ω R ⋆ ( C ) and H z ⊛ ω R ( C ) ⊂ H z ⊛ ω O ( C ) ⊂ H z ⊛ ω O ⋆ ( C ) ⊂ H z ⊛ ω R ⋆ ( C ) If H is an injectiv e Ho pf alg ebra, then the uppe r chain is deﬁned, if H is a pr o jective Hopf algebra , then the low er chain is deﬁned, and if H is a rigid stereotype Ho pf algebr a, then b oth chains are deﬁned (and, certainly , they coincide up to is omorphisms). Theorem 7.6 co rrectly deﬁne only the ﬁrst link in the ﬁrst chain and the last link in the seco nd chain. In all conscie nce, to give accurate deﬁnition for all links we should form ulate three mo re analogous theorems. T o av oid thos e tro ubles we can either simply sa y that the other links ar e deﬁned b y ana logy (with replacing, if necess ary , ⊙ by ⊛ , a nd R by O ). Or we can unite all those four theorems (the one alre ady prov en and three not yet formulated) into the following quite bulky pr opos ition: Theorem 7.7. L et — F denote one of the two Hopf algebr as: R ( C ) or O ( C ) , — H b e an arbitr ary inje ctive ster e otyp e Hopf algebr a, — ( z , ω ) b e a qu ant um p air in H with p ar ameter q ∈ C × . Then 7 (a) the tensor pr o duct H ⊙ F p ossesses a unique structur e of inje ctive Hopf algebr a with algebr aic op er ations, deﬁne d by formulas: multiplic ation: a ⊙ t k · b ⊙ t l = a · ( M ⋆ ω ) k ( b ) ⊙ t k + l (7.35) 7 Here again θ is the is omorphism of functors (1.13), and ( k )! q the quantu m factorial deﬁned in (7.2). 116 unit: 1 H ⊙R ( C ) = 1 H ⊙ 1 R ( C ) (7.36) c omu ltipli c ation: κ ( a ⊙ t k ) = k X i =0  k i  q · θ   1 H ⊙ M i z  ( κ H ( a )) ⊙ t i ⊙ t k − i  = (7.3 7) = X ( a ) k X i =0  k i  q · a ′ ⊙ t i ⊙ ( z i · a ′′ ) ⊙ t k − i c oun it: ε ( a ⊙ t k ) = ( ε H ( a ) , k = 0 0 , k > 0 (7.38) antip o de: σ ( a ⊙ t k ) = ( − 1) k · q − k ( k +1) 2 · z − k · ( M ⋆ ω ) k ( σ H ( a )) ⊙ t k (7.39) H ⊙ F with such a st ructur e of Hopf algebr a is denote d by H z ⊙ ω F ; (b) the tensor pr o duct H ⋆ ⊛ F ⋆ p ossesses a unique structu r e of pr oje ctive Hopf algebr a with t he algebr aic op er ations, deﬁne d by formulas: multiplic ation: α ⊛ τ k ∗ β ⊛ τ l = α · ( M ⋆ z ) k ( β ) ⊛ τ k + l (7.40) unit: 1 H ⋆ ⊛ R ( C ) = 1 H ⋆ ⊛ 1 R ⋆ ( C ) (7.41) c omu ltipli c ation: κ ( α ⊛ τ k ) = k X i =0  k i  q · θ   id H ⋆ ⊛ M i ω  ( κ H ⋆ ( α )) ⊛ τ i ⊛ τ k − i  = (7.42) = X ( α ) k X i =0  k i  q · α ′ ⊛ τ i ⊛ ( ω i · α ′′ ) ⊛ τ k − i c oun it: ε ( α ⊛ τ k ) = ( ε H ⋆ ( α ) , k = 0 0 , k > 0 (7.43) antip o de: σ ( α ⊛ τ k ) = ( − 1) k · q − k ( k +1) 2 · ω − k ∗ ( M ⋆ z ) k ( σ H ⋆ ( α )) ⊛ τ k (7.44) H ⋆ ⊛ F ⋆ with s u ch a stru ctur e of Hopf algebr a is denote d by H ⋆ ω ⊛ z F ⋆ ; (c) the biline ar form * X k ∈ N u k ⊙ t k , X k ∈ N α k ⊛ τ k + = X k ∈ N h u k , α k i · ( k )! q (7.45) turns H z ⊙ ω F and H ⋆ ω ⊛ z F ⋆ into dual p air of ster e otyp e Hopf algebr as. If, under al l other assumptions (exc ept inje ctivity), H is a pr oje ctive ster e otyp e Hopf algebr a, t hen (a) the tensor pr o duct H ⊛ F has a u n ique struct u r e of pr oje ctive ster e otyp e Hopf algebr a with algebr aic op er ations deﬁne d by formulas (7.35) - (7.39) , but with r eplacing ⊙ by ⊛ ; H ⊛ F with such a str u ctur e of Hopf algebr a is denote d by H z ⊛ ω F ; (b) the tensor pr o duct H ⋆ ⊙ F ⋆ has a u nique st ructur e of inje ctive ster e otyp e Hopf algebr a with algebr aic op er ations deﬁne d by formulas (7.40) - (7.44) , but with r eplacing ⊛ by ⊙ ; H ⋆ ⊙ F ⋆ with such a structur e of H opf algebr a is denote d by H ⋆ ω ⊙ z F ⋆ (c) the biline ar form * X k ∈ N u k ⊛ t k , X k ∈ N α k ⊙ τ k + = X k ∈ N h u k , α k i · ( k )! q (7.46) § 7. APPE NDIX: HOLO MORPHIC REFLEXIVITY OF THE QUANTUM GR OUP ‘ AZ + B ’ 117 turns H z ⊛ ω F and H ⋆ ω ⊙ z F ⋆ into a dual p air of ster e otyp e Hopf algebr as. As we hav e alr eady told this is prov ed by analogy with Theor em 7 .6. Prop osition 7.2. L et H b e an inje ctive Hopf algebr a, and ( z , ω ) a qu ant um p air in H with the p ar ameter q ∈ C × . Then the rules a ⊙ t k 7→ a ⊙ t k 7→ a ⊙ τ k 7→ a ⊙ τ k ( k ∈ N , a ∈ H ) uniquely deﬁne a chain of (c ontinuous) homomorphisms of inje ctive ster e otyp e Hopf algebr as: H z ⊙ ω R ( C ) → H z ⊙ ω O ( C ) → H z ⊙ ω O ( C ) ⋆ → H z ⊙ ω R ( C ) ⋆ Prop osition 7.3. L et H b e a pr oje ctive Hopf algebr a and ( z , ω ) a quantum p air in H with the p ar ameter q ∈ C × . Then the rules a ⊛ t k 7→ a ⊛ t k 7→ a ⊛ τ k 7→ a ⊛ τ k ( k ∈ N , a ∈ H ) uniquely deﬁne a chain of (c ontinuous) homomorphisms of pr oje ctive ster e otyp e Hopf algebr as: H z ⊛ ω R ( C ) → H z ⊛ ω O ( C ) → H z ⊛ ω O ( C ) ⋆ → H z ⊛ ω R ( C ) ⋆ (c) Quan t um group ‘ az + b ′ = R q ( C × ⋉ C ) Here we show that the quantum group ‘ az + b ’ (deﬁned in [48, 4 0, 47, 28]) is a spe cial case of the construction describ ed in Theorem 7.6. The g roup C × ⋉ C of aﬃne transformation of a complex plane. The g roup of aﬃne transfor - mations of the complex plane, often deno ted as ‘ az + b ’, fr om the a lgebraic p oin t of view is a semidir ect pro duct C × ⋉ C of complex cir cle C × and complex pla ne C , where C × acts on C by usual mult iplication. In other word, C × ⋉ C is a Car tesian pr oduct C × × C with a lgebraic op erations m ultiplication: ( a, x ) · ( b, y ) = ( ab, xb + y ) ( a, b ∈ C × , x, y ∈ C ) unit: 1 C × ⋉C = (1 , 0) inv erse element: ( a, x ) − 1 =  1 a , − x a  ( a ∈ C × , x ∈ C ) Clearly , this is a connected Stein group. Mo reov er, C × ⋉C is an a lgebraic group, since it can b e repre sen ted as a linear gro up by matrices o f the form ( a, x ) =  a 0 x 1  ( a ∈ C × , x ∈ C ) (and multiplication, unit a nd inverse element b ecome us ual op erations with ma trices). Stereot yp e alge b ras R ( C × ⋉ C ) and R ⋆ ( C × ⋉ C ) . By symbo l R ( C × ⋉ C ) we, as usual, deno te the algebra of p olynomia ls o n an algebra ic gro up C × ⋉ C . According to the genera l approa c h o f § 3(b), we endow the space R ( C × ⋉ C ) with the stronges t lo cally conv ex topo logy . The dual space of curr en ts R ⋆ ( C × ⋉ C ) is a n algebra with r espect to the usual conv olution of functionals (3.14). Like any other dual space to a stereo t yp e space, it is endow ed with the to polog y of uniform conv erge nce on c ompact sets in R ( C × ⋉ C ). In this cas e this is equiv alent to the R ( C × ⋉ C )-weak topo logy . 118 Recall that by z n and t k we denote bas is monomials in the spac es R ( C × ) a nd R ( C ) (we deﬁned them by formulas (3.30) and (3.52)). In accordance wit h the common notation (1.24), it is reasonable to denote the basis monomials in the space of functions R ⋆ ( C × ⋉ C ) by symbo l z n ⊡ t k : ( z n ⊡ t k )( a, x ) := a n · x k , a ∈ C × , x ∈ C Similarly , extending the old notations ζ n and τ k from (3.31) a nd (3.5 9), we denote by ζ n ∗  τ k the functional on R ( C × ⋉ C ) of taking the n -th co eﬃcient of Laur en t ser ies with resp ect to the ﬁrst v a riable and a t the same time the k -th deriv ative in the p oint (1 , 0) with r espect to the s econd v ariable: ζ n ∗  τ k ( u ) = Z 1 0 e − 2 π int d k d x k u ( x, e 2 π it )    x =0 d t = d k d x k Z 1 0 e − 2 π int u ( x, e 2 π it ) d t    x =0 (7.47) Prop osition 7.4. 1) The functions { z n ⊡ t k ; n ∈ Z , k ∈ N } form an algebr aic b asis in t he sp ac e R ( C × ⋉C ) of p olynomia ls on C × ⋉ C : every p olynomial u ∈ R ( C × ⋉ C ) is uniquely de c omp ose d int o t he series u = X k ∈ N ,n ∈ Z u n,k · z n ⊡ t k , card { ( n, k ) : u n,k 6 = 0 } < ∞ , (7.48) wher e the c o eﬃcients c an b e c ompute d by formula u n,k = 1 k ! · ζ n ∗  τ k ( u ) (7.49) The c orr esp ondenc e u ↔ { u n,k ; n ∈ Z , k ∈ N } establishes an isomorphism of top olo gic al ve ctor sp ac es R ( C × ⋉ C ) ∼ = C Z × N 2) The functionals { ζ n ∗  τ k ; n ∈ Z , k ∈ N } form a b asis in t he ster e otyp e sp ac e R ⋆ ( C × ⋉ C ) : every functional α ∈ R ⋆ ( C × ⋉ C ) is un iquely de c omp ose d into a (c onver ging in t he sp ac e R ⋆ ( C × ⋉ C ) ) series α = X k ∈ N ,n ∈ Z α n,k · ζ n ∗  τ k , (7.50 ) wher e the c o eﬃcients c an b e c ompute d by formula α n,k = 1 k ! · α ( z n ⊡ t k ) (7.51) The c orr esp ondenc e α ↔ { α n,k ; n ∈ Z , k ∈ N } establishes an isomorphism of top olo gic al ve ctor sp ac es R ⋆ ( C × ⋉ C ) ∼ = C Z × N 3) The b ases { z n ⊡ t k ; n ∈ Z , k ∈ N } and { ζ n ∗  τ k ; n ∈ Z , k ∈ N } ar e dual to e ach other up to the c onstant k ! h z m ⊡ t k , ζ n ∗ τ l i = h z m , ζ n i · h t k , τ l i = ( 0 , ( m, k ) 6 = ( n, l ) k ! , ( m, k ) = ( n, l ) (7.52) and the action of functionals α ∈ R ⋆ ( C × ⋉ C ) on p olynomials u ∈ R ( C × ⋉ C ) is describ e d by the formula h u, α i = X n ∈ Z ,k ∈ N u n,k · α n,k · k ! (7.53) Remark 7.1. The functional ζ n ∗  τ k can b e repr esen ted as the conv olution o f tw o comp onents ζ n and τ k (and the o rder of their multiplication b ecomes impo rtan t here ): if we denote by Z n the functional of taking the n -th co eﬃcient of Laur en t series with resp ect to the ﬁrst v a riable in the p oin t (1 , 0), Z n ( u ) = ζ n ∗  τ 0 ( u ) = 1 2 π i Z | z | =1 u ( z , 0 ) z n +1 d z = Z 1 0 e − 2 π int u ( e 2 π it , 0) d t, u ∈ R ( C × ⋉ C ) , (7.54) § 7. APPE NDIX: HOLO MORPHIC REFLEXIVITY OF THE QUANTUM GR OUP ‘ AZ + B ’ 119 and by T k the functional of taking the k -th deriv ative with resp ect to the second v ariable in the p oint (1 , 0), T k ( u ) = ζ 0 ∗  τ k ( u ) = d k d x k u (1 , x )    x =0 , u ∈ R ( C × ⋉ C ) , (7.55) then the following equa lities hold: Z n ∗ T k = ζ n ∗  τ k , T k ∗ Z n = Z n − k ∗ T k = ζ n − k ∗  τ k (7.56) F o r the pro of we can use formulas (3.1 5): ﬁrst, δ a ∗ T k = a · T k ⇓ ( δ ( e 2 πit , 0) ∗ T k )( u ) = (( e 2 π it , 0) · T k )( u ) = T k ( u · ( e 2 π it , 0)) = = d k d x k u  ( e 2 π it , 0) · (1 , x )     x =0 = d k d x k u ( e 2 π it , x )    x =0 (7.57) ⇓ ( Z n ∗ T k )( u ) =  Z 1 0 e − 2 π int · δ ( e 2 πit , 0) d t ∗ T k  ( u ) =  Z 1 0 e − 2 π int · δ ( e 2 πit , 0) ∗ T k d t  ( u ) = = Z 1 0  e − 2 π int · δ ( e 2 πit , 0) ∗ T k  ( u ) d t = Z 1 0 e − 2 π int · d k d x k u ( e 2 π it , x )    x =0 d t And, seco nd, T k ∗ δ a = T k · a ⇓ ( T k ∗ δ ( e 2 πit , 0) )( u ) = ( T k · ( e 2 π it , 0))( u ) = T k (( e 2 π it , 0) · u ) = d k d x k u  (1 , x ) · ( e 2 π it , 0)     x =0 = = d k d x k u ( e 2 π it , x · e 2 π it )    x =0 = ( e 2 π it ) k · d k d y k u ( e 2 π it , y )    y =0 = (7.57) = e 2 π ikt · ( δ ( e 2 πit , 0) ∗ T k )( u ) ⇓ T k ∗ δ ( e 2 πit , 0) = e 2 π ikt · δ ( e 2 πit , 0) ∗ T k (7.58) ⇓ T k ∗ Z n = T k ∗ Z 1 0 e − 2 π int · δ ( e 2 πit , 0) d t = Z 1 0 e − 2 π int · T k ∗ δ ( e 2 πit , 0) d t = (7.58) = = Z 1 0 e − 2 π int · e 2 π ikt · δ ( e 2 πit , 0) ∗ T k d t = Z 1 0 e − 2 π i ( n − k ) t · δ ( e 2 πit , 0) ∗ T k d t = = Z 1 0 e − 2 π i ( n − k ) t · δ ( e 2 πit , 0) d t ∗ T k = Z n − k ∗ T k 120 Stereot yp e alge b ras O ( C × ⋉ C ) and O ⋆ ( C × ⋉ C ) . As always, we endow the alg ebra O ( C × ⋉ C ) of holomorphic functions on C × ⋉ C with the to polog y of uniform conv erg ence on co mpact sets in C × ⋉ C . Its dual algebr a O ⋆ ( C × ⋉ C ) is endow ed with the top olog y of uniform conv ergence on compact sets in O ( C × ⋉ C ), and the multiplication there is the usua l conv olution (3.14). Prop osition 7 .5. 1) The fun ctions { z n ⊡ t k ; n ∈ Z , k ∈ N } form a b asis in t he ster e otyp e sp ac e O ( C × ⋉ C ) of holo morphic functions on C × ⋉ C : every fun ct ion u ∈ O ( C × ⋉ C ) c an b e uniquely re pr esente d as a sum of a (c onver ging in O ( C × ⋉ C ) ) series u = X k ∈ N ,n ∈ Z u n,k · z n ⊡ t k , ∀ C > 0 X k ∈ N ,n ∈ Z | u k,n | · C k + | n | < ∞ , (7.5 9) wher e the c o eﬃcients (c ontinuously dep end on u and) c an b e c ompute d by formula u n,k = 1 k ! · ζ n ∗  τ k ( u ) (7.60) The top olo gy of O ( C × ⋉ C ) c an b e describ e d by seminorms: || u || C = X k ∈ N ,n ∈ Z | u k,n | · C k + | n | , C > 1 (7.61) 2) The functionals { ζ n ∗  τ k ; n ∈ Z , k ∈ N } form a b asis in the sp ac e O ⋆ ( C × ⋉ C ) : every functional α ∈ O ⋆ ( C × ⋉ C ) c an b e uniquely r epr esente d as a su m of a (c onver ging in O ⋆ ( C × ⋉ C ) ) series α = X k ∈ N ,n ∈ Z α n,k · ζ n ∗  τ k , (7.62 ) wher e the c o eﬃcients (c ontinuously dep end on α and) c an b e c ompute d by formula α n,k = 1 k ! · α ( z n ⊡ t k ) (7.63) The top olo gy of t he sp ac e O ⋆ ( C × ⋉ C ) c an b e describ e d by seminorms: ||| α ||| r = X n ∈ Z ,k ∈ N r n,k · | α n,k | · k ! , r n,k > 0 : ∀ C > 1 X n ∈ Z ,k ∈ N r k,n · C k + | n | < ∞ (7.64) 3) The b ases { z n ⊡ t k ; n ∈ Z , k ∈ N } and { ζ n ∗  τ k ; n ∈ Z , k ∈ N } ar e dual to e ach other up to the c onst ant k ! : h z m ⊡ t k , ζ n ∗ τ l i = h z m , ζ n i · h t k , τ l i = ( 0 , ( m, k ) 6 = ( n, l ) k ! , ( m, k ) = ( n, l ) (7.65) and the action of funct ionals α ∈ O ⋆ ( C × ⋉ C ) on fun ct ions u ∈ O ( C × ⋉ C ) is describ e d by formula h u, α i = X n ∈ Z ,k ∈ N u n,k · α n,k · k ! (7.66) R ( C × ⋉ C ) , R ⋆ ( C × ⋉ C ) , O ( C × ⋉ C ) and O ⋆ ( C × ⋉ C ) as Hopf algebras. The algebras R ( C × ⋉ C ) and O ( C × ⋉ C ), b eing standard functiona l alg ebras on g roups, are endow ed with the natur al structure of Hopf algebra s (we noted this genera l fa ct in Theorems 3.2 and 3.1). The following prop ositions des cribe the structure of these Hopf algebr as. Prop osition 7 .6. The algebr a R ( C × ⋉ C ) (r esp., algebr a O ( C × ⋉ C ) ) is a nucle ar Hopf-Br auner (r esp., Hopf-F r ´ echet) algebr a with the algebr aic op er ations deﬁne d on b asis elements z n ⊡ t k by formulas z m ⊡ t k · z n ⊡ t l = z m + n ⊡ t k + l 1 R ( C × ⋉C ) = z 0 t 0 (7.67) § 7. APPE NDIX: HOLO MORPHIC REFLEXIVITY OF THE QUANTUM GR OUP ‘ AZ + B ’ 121 κ ( z n ⊡ t k ) = k X i =0  k i  · z n t i ⊙ z n + i t k − i ε ( z n ⊡ t k ) = ( 0 , ( n, k ) 6 = (0 , 0 ) 1 , ( n, k ) = (0 , 0 ) (7.68) σ ( z n ⊡ t k ) = ( − 1) k z − k − n t k (7.69) As a c or ol lary, the gener al formula of multiplic ation in R ( C × ⋉ C ) is as fol lows: u · v = X n ∈ Z ,k ∈ N k X j =0 X i ∈ Z u i,j · v n − i,k − j ! · z n ⊡ t k . (7.70) Pr o of. The comultip lication, co unit and antipo de can b e computed by formulas (3.5)-(3.7). F or instance, the comultiplication: e κ ( z n ⊡ t k )  ( a, x ) , ( b, y )  = ( z n ⊡ t k )  ( a, x ) · ( b, y )  = ( z n ⊡ t k )( ab, xb + y ) = ( ab ) n ( xb + y ) k = = a n b n k X i =0  k i  x i b i y k − i = k X i =0  k i  a n x i b n + i y k − i = k X i =0  k i  ( z n ⊡ t i ) ⊡ ( z n + i ⊡ t k − i )  ( a, x ) , ( b, y )  ⇓ e κ ( z n ⊡ t k ) = k X i =0  k i  · ( z n ⊡ t i ) ⊡ ( z n + i ⊡ t k − i ) ⇓ κ ( z n ⊡ t k ) = k X i =0  k i  · z n ⊡ t i ⊙ z n + i ⊡ t k − i Prop osition 7. 7 . The algebr a R ⋆ ( C × ⋉ C ) (r esp., algebr a O ⋆ ( C × ⋉ C ) ) is a nucle ar Hopf-F r ´ echet (r esp., Hopf-Br auner) algebr a with the algebr aic op er ations deﬁne d on b asis elements ζ n ∗  τ k by formulas ( ζ m ∗  τ k ) ∗ ( ζ n ∗  τ l ) = ( ζ m ∗  τ k + l , m + k = n 0 , m + k 6 = n 1 R ⋆ ( C × ⋉C ) = X n ∈ Z ζ n ∗  τ 0 (7.71) κ ( ζ n ∗  τ k ) = X m ∈ Z k X i =0  k i  · ζ m ∗  τ i ⊛ ζ n − m ∗  τ k − i ε ( ζ n ∗  τ k ) = ( 1 , ( n, k ) = (0 , 0) 0 , ( n, k ) 6 = (0 , 0) (7.72) σ ( ζ n ∗  τ k ) = ( − 1) k · ζ − n − k ∗  τ k (7.73) As a c or ol lary, the gener al formula of multiplic ation in R ⋆ ( C × ⋉ C ) is as fol lows: α ∗ β = X n ∈ Z ,k ∈ N  k X j =0 α n,j · β n + j,k − j  · ζ n ∗  τ k . (7.74 ) Pr o of. All those f ormulas app ear as duals of (7.67)-(7.69). F or example, the com ultiplication is computed as follows: h u ⊙ v , κ ( ζ n ∗  τ k ) i = h u · v , ζ n ∗  τ k i = * X r ∈ Z ,l ∈ N l X i =0 X m ∈ Z u m,i · v r − m,l − i ! · z r t l , ζ n ∗  τ k + = = k ! · k X i =0 X m ∈ Z u m,i · v n − m,k − i = k ! · k X i =0 X m ∈ Z 1 i ! h u, ζ m ∗  τ i i · 1 ( k − i )! h v , ζ n − m ∗  τ k − i i = 122 = X m ∈ Z k X i =0  k i  · h u ⊙ v , ζ m ∗  τ i ⊛ ζ n − m ∗  τ k − i i = * u ⊙ v, X m ∈ Z k X i =0  k i  · ζ m ∗  τ i ⊛ ζ n − m ∗  τ k − i + ⇓ κ ( ζ n ∗  τ k ) = X m ∈ Z k X i =0  k i  · ζ m ∗  τ i ⊛ ζ n − m ∗  τ k − i Hopf algebras R q ( C × ⋉ C ) , R ⋆ q ( C × ⋉ C ) , O q ( C × ⋉ C ) , O ⋆ q ( C × ⋉ C ) . The q uan tum gr oup ‘ az + b ’ can be deﬁned as Hopf a lgebra R ( C × ⋉ C ), wher e the a lgebraic op erations a re deformed in so me sp ecial wa y . W e descr ibe her e this deformatio n, a nd together with the a lgebra R ( C × ⋉ C ) we s hall cons ider the algebra O ( C × ⋉ C ). Both these c onstructions will b e useful b elow in Theor em 7.8. The constructio ns starts with the choice o f a constant q ∈ C × . Prop osition 7.8. On the ster e otyp e sp ac e R ( C × ⋉ C ) (r esp e ctively, O ( C × ⋉ C ) ) ther e exists a u nique structur e of rigid ster e otyp e Hopf algebr a with the algebr aic op er ations deﬁne d on b asis elements z n ⊡ t k by formulas z m ⊡ t k · z n ⊡ t l = q kn · z m + n ⊡ t k + l 1 R ( C × ⋉C ) = z 0 t 0 (7.75) κ ( z n ⊡ t k ) = k X i =0  k i  q · z n ⊡ t i ⊙ z n + i ⊡ t k − i ε ( z n ⊡ t k ) = ( 0 , ( n, k ) 6 = (0 , 0 ) 1 , ( n, k ) = (0 , 0 ) (7.76) σ ( z n ⊡ t k ) = ( − 1) k · q − k ( k +1) 2 − kn · z − k − n ⊡ t k (7.77) The sp ac e R ( C × ⋉ C ) (r esp e ctively, O ( C × ⋉ C ) ) with such a structur e of Hopf algebr a is deﬁne d by R q ( C × ⋉ C ) (r esp e ctively, by O q ( C × ⋉ C ) ). Besides this, 1) the gener al formula of multiplic ation in R q ( C × ⋉ C ) (r esp e ctively, in O q ( C × ⋉ C ) ) has the form u · v = X n ∈ Z ,k ∈ N k X i =0 X m ∈ Z q i ( n − m ) · u m,i · v n − m,k − i ! · z n ⊡ t k . (7.78) 2) The mapping z n ⊡ t k 7→ z n ⊙ t k establishes an isomorphism b etwe en R q ( C × ⋉ C ) (r esp e ctively, O q ( C × ⋉ C ) ) and the Hopf algebr a of skew p olynomials (en tir e functions) with c o eﬃcients in R ( C × ) (r esp e ctively, in O ( C × ) ) with r esp e ct to the quantum p air ( z , δ q ) fr om Pr op osition 7.1: R q ( C × ⋉ C ) ∼ = R ( C × ) z ⊙ δ q R ( C ) O q ( C × ⋉ C ) ∼ = O ( C × ) z ⊙ δ q O ( C ) ! (7.79) 3) F or q = 1 the Hopf algebr a R q ( C × ⋉ C ) (r esp e ctively, O q ( C × ⋉ C ) ) turns int o the Hopf algebr a R ( C × ⋉ C ) (r esp e ctively, O ( C × ⋉ C ) ) with the structu r e of Hopf algebr a describ e d on p age 117: R ( C × ⋉ C ) = R 1 ( C × ⋉ C ) O ( C × ⋉ C ) = O 1 ( C × ⋉ C ) ! § 7. APPE NDIX: HOLO MORPHIC REFLEXIVITY OF THE QUANTUM GR OUP ‘ AZ + B ’ 123 Pr o of. Here ev erything starts from for m ulas (7.79): the mapping z n ⊡ t k 7→ z n ⊙ t k establishes an isomorphism of stereo t yp e spaces R q ( C × ⋉ C ) ∼ = R ( C × ) ⊙ R ( C ) (this is exactly the isomor phism, which for general case is de scribed by the identit y (3.9)). This iso- morphism induces on R ( C × ⋉ C ) the structure of r igid Hopf algebr a from R ( C × ) z ⊙ δ q R ( C ) (wher e this structure is deﬁned by formulas (7.14)-(7.1 8 )). In this is omorphism the formulas (7.14)-(7.18) tur n into formulas (7.75)-(7.7 7 ). T o derive them we need to use the ﬁrst formula in (7 .12): M ⋆ δ q ( z ) = q · z This implies ( M ⋆ δ q ) k ( z n ) = q kn · z n Then, for instance, the formula of multiplication (7 .75 ) is derived from (7.14) as follows: z m ⊡ t k · z n ⊙ t l = z m · ( M ⋆ δ q ) k ( z n ) · t k + l = z m · q kn · z n ⊙ t k + l = q kn · z k + n ⊙ t k + l The rema ining formulas ar e deduced b y a nalogy . The gener al fo rm ula for multiplication (7.78) follows from (7.75): u · v =   X m ∈ Z ,k ∈ N u m,k · z m ⊡ t k   ·   X n ∈ Z ,l ∈ N v n,l · z n ⊡ t l   = X m ∈ Z ,k ∈ N X n ∈ Z ,l ∈ N u m,k · v n,l · z m ⊡ t k · z n ⊡ t l = = X m ∈ Z ,k ∈ N X n ∈ Z ,l ∈ N u m,k · v n,l · q kn · z m + n t k + l =  n = r − m l = s − k  = = X r ∈ Z ,s ∈ N X m ∈ Z ,k ∈ N q k ( r − m ) · u m,k · v r − m,s − k · z m + n t k + l It remains to add that for q = 1 the formulas (7.75)-(7.77) turn into formulas (7.67)-(7 .69) , so the Hopf alg ebra R q ( C × ⋉ C ) (resp ectively , O q ( C × ⋉ C )) turns into Hopf alge bra R ( C × ⋉ C ) (re spectively , O ( C × ⋉ C )). Prop osition 7.9. On the ster e otyp e sp ac e R ⋆ ( C × ⋉ C ) (re sp e ct ively , O ⋆ ( C × ⋉ C ) ) ther e exists a un ique structur e of rigid s ter e otyp e H opf algebr a with algebr aic op er ations deﬁne d on b asis elements ζ n ∗  τ k by formulas ( ζ m ∗  τ k ) ∗ ( ζ n ∗  τ l ) = ( ζ m ∗  τ k + l , m = n − k 0 , m 6 = n − k 1 R ⋆ ( C × ⋉C ) = X n ∈ Z ζ n ∗  τ 0 (7.80) κ ( ζ n ∗  τ k ) = X m ∈ Z k X i =0  k i  q · q i ( n − m ) · ζ m ∗  τ i ⊛ ζ n − m ∗  τ k − i ε ( ζ n ∗  τ k ) = ( 1 , k = 0 0 , k > 0 (7.81) σ ( ζ n ∗  τ k ) = ( − 1) k · q − k ( k +1) 2 + k ( n + k ) ζ − n − k ∗  τ k (7.82) The sp ac e R ⋆ ( C × ⋉ C ) (r esp e ctively, O ⋆ ( C × ⋉ C ) ) with t his struct ur e of H opf algebr a is denote d by R ⋆ q ( C × ⋉ C ) (r esp e ctively, O ⋆ q ( C × ⋉ C ) ). Besides this, 1) the gener al formula for multiplic ation in R ⋆ q ( C × ⋉ C ) (r esp e ctively, in O ⋆ q ( C × ⋉ C ) ) has the form α ∗ β = X n ∈ Z ,k ∈ N  k X i =0 α n,i · β n + i,k − i  · ζ n ∗  τ k . (7.83 ) 124 2) the mapping ζ n ∗  τ k 7→ ζ n ⊛ τ k establishes an isomorphism b etwe en R ⋆ q ( C × ⋉ C ) (r esp e ctively, O ⋆ q ( C × ⋉ C ) ) and the Hopf algebr a of skew p ower series (r esp e ctively, analytic functionals) with c o eﬃcients in R ⋆ ( C × ) ( re sp e ct ively , in O ⋆ ( C × ) ) with r esp e ct t o the quant u m p air ( δ q , z ) fr om Pr op osition 7.1 : R ⋆ q ( C × ⋉ C ) ∼ = R ⋆ ( C × ) δ q ⊙ z R ⋆ ( C ) , O ⋆ q ( C × ⋉ C ) ∼ = O ⋆ ( C × ) δ q ⊙ z O ⋆ ( C ) (7.84) 3) for q = 1 the Hopf algebr a R ⋆ q ( C × ⋉ C ) (r esp e ctively, O ⋆ q ( C × ⋉ C ) ) turns into the Hopf algebr a R ⋆ ( C × ⋉ C ) (r esp e ctively, O ⋆ ( C × ⋉ C ) ) with the st r u ctur e of Hopf algebr a deﬁne d on p age 117: R ⋆ ( C × ⋉ C ) = R ⋆ 1 ( C × ⋉ C ) O ⋆ ( C × ⋉ C ) = O ⋆ 1 ( C × ⋉ C ) ! Pr o of. Here again everything is bas ed on fo rm ulas (7.84): the mapping ζ n ∗  τ k 7→ ζ n ⊛ τ k establishes an isomorphism of stereo t yp e spaces R ⋆ q ( C × ⋉ C ) ∼ = R ⋆ ( C × ) ⊛ R ⋆ ( C ) (this is the dual iso morphism for (3.9)). This isomor phism induces on R ⋆ ( C × ⋉ C ) a structure of r igid Hopf alg ebra fro m R ⋆ ( C × ) δ q ⊛ z R ⋆ ( C ) (where this structur e is deﬁned by formulas (7 .20)-(7.24)). Under this is omorphism formulas (7.20)-(7.24) turn into for m ulas (7.80)-(7.82). T o derive them we can use t wo formulas: M ⋆ z ( ζ n ) = ζ n − 1 , δ q ∗ ζ n = q n · ζ n (7.85) The ﬁrst of them (it will b e useful in pr o ving for m ulas for mu ltiplication a nd antipo de), is deduced a s follows: h u, M ⋆ z ( ζ n ) i = h z · u, ζ n i = * z · X m ∈ Z u m · z m , ζ n + = * X m ∈ Z u m · z m +1 , ζ n + = = * X m ∈ Z u m − 1 · z m , ζ n + = u n − 1 = h u, ζ n − 1 i And the seco nd one (it will b e useful in pr o ving the formula for comultiplication) a s follows: δ q ∗ ζ n = (7.13) = X m ∈ Z q m · ζ m ∗ ζ n = (3.41) = q n · ζ n R q ( C × ⋉ C ) as an algebra with generators and de ﬁning rel ations. It remains to explain why the constructed algebr a R q ( C × ⋉ C ) indeed can b e iden tiﬁed with the quantum gro up ‘ a z + b ’, i.e. with the Hopf algebr a which is usua lly denoted a s ‘ az + b ’ (see [48 , 40, 47, 2 8]). F ormally ‘ a z + b ’ is deﬁned a s the Hopf algebra with three generator s t , z , z − 1 and the deﬁning rela tions t · z = q · z · t z · z − 1 = 1 1 = z − 1 · z (7.86) κ ( t ) = t ⊗ 1 + z ⊗ t, κ ( z ) = z ⊗ z , κ ( z − 1 ) = z − 1 ⊗ z − 1 (7.87) ε ( t ) = 0 , ε ( z ) = 1 , ε ( z − 1 ) = 0 (7.88) σ ( t ) = − t · z − 1 , σ ( z ) = z − 1 , σ ( z − 1 ) = z , (7.89 ) § 7. APPE NDIX: HOLO MORPHIC REFLEXIVITY OF THE QUANTUM GR OUP ‘ AZ + B ’ 125 Prop osition 7.10. The mapping z 7→ z ⊡ 1 , z − 1 7→ z − 1 ⊡ 1 , t 7→ 1 ⊡ t is uniquely extende d to an isomorphism b etwe en Hopf algebr as ‘ az + b ’ and R q ( C × ⋉ C ) . Pr o of. F r om (7.75) it follows tha t in R q ( C × ⋉ C ) the following identities hold 1 ⊡ t · z ⊡ 1 = q · z ⊡ 1 · 1 ⊡ t, z ⊡ 1 · z − 1 ⊡ 1 = 1 , 1 = z − 1 ⊡ 1 · z ⊡ 1 in which one can recog nize formulas (7.8 6), trans formed by our mapping. This means that our ma pping can b e uniquely extended to some homomorphism of algebra s ϕ : ‘ az + b ′ → R q ( C × ⋉ C ). This homo- morphism is a bijection since it turns the algebr aic basis { z n t k ; n ∈ Z , k ∈ N } in alg ebra ‘ az + b ’ into the algebra ic basis { z n ⊡ t k ; n ∈ Z , k ∈ N } in algebr a R q ( C × ⋉ C ). Thu s, ϕ is a n isomorphism of algebra s. Note then that ϕ preserves the com ultiplication on generators: ( ϕ ⊗ ϕ )( κ ( t )) = ( ϕ ⊗ ϕ )( t ⊗ 1 + z ⊗ t ) = 1 ⊡ t ⊗ 1 ⊡ 1 + z ⊡ 1 ⊗ 1 ⊡ t = κ (1 ⊡ t ) = κ ( ϕ ( t )) and similarly , ( ϕ ⊗ ϕ )( κ ( z )) = κ ( ϕ ( z )) , ( ϕ ⊗ ϕ )( κ ( z − 1 )) = κ ( ϕ ( z − 1 )) Since the comultiplication, like ϕ , is a homomorphism of algebra s, this implies that the same f ormulas are true for the arguments of the form z n t k , and th us f or all element s of algebra ‘ az + b ’. W e obtain that ϕ pre- serves the comultiplication (o n a ll elements). Similarly it is prov ed that ϕ preser v es counit and antipo de (here we need to use the fact that the co unit is a ho momorphism, and antipo de an antihomomorphism). Hence, ϕ is an iso morphism of Hopf a lgebras. (d) Reﬂexivit y of R q ( C × ⋉ C ) Reﬂexivity diagram for R q ( C × ⋉ C ) . Theorem 7.8. F or any q ∈ C × the rigid Hopf algebr a R q ( C × ⋉ C ) is holomorphic al ly r eﬂexive, and 1) for | q | = 1 its r eﬂexivity diagr am has the form R q ( C × ⋉ C ) ∼ = R ( C × ) z ⊙ δ q R ( C )  ♥ / / O ( C × ) z ⊙ δ q O ( C ) ∼ = O q ( C × ⋉ C ) _ ⋆   _ ⋆ O O R ⋆ q ( C × ⋉ C ) ∼ = R ⋆ ( C × ) δ q ⊙ z R ⋆ ( C )  ♥ o o O ⋆ ( C × ) δ q ⊙ z O ⋆ ( C ) ∼ = O ⋆ q ( C × ⋉ C ) (7.90) 2) for | q | 6 = 1 the form: R q ( C × ⋉ C ) ∼ = R ( C × ) z ⊙ δ q R ( C )  ♥ / / O ( C × ) z ⊙ δ q R ⋆ ( C ) _ ⋆   _ ⋆ O O R ⋆ q ( C × ⋉ C ) ∼ = R ⋆ ( C × ) δ q ⊙ z R ⋆ ( C )  ♥ o o O ⋆ ( C × ) δ q ⊙ z R ( C ) (7.91) 126 F o r symmetry o ne ca n note her e that all the multipliers here – R ( C ), R ⋆ ( C ), R ( C × ), R ⋆ ( C × ), O ( C × ), O ⋆ ( C × ) – are nuclear spaces and a ppear only in pair F r´ echet-F r´ echet a nd Brauner-Br auner. So if in diagrams (7.90)-(7.9 1 ) a ll (or so me of ) the injective tensor pro ducts ⊙ a re replaced by pro jective tensor pro ducts ⊛ , then we shall obta in iso morphic diagr ams. The rest of this section is devoted to the pro of o f Theo rem 7 .8. W e carry out it in three steps. R q ( C × ⋉ C ) ♥ → O q ( C × ⋉ C ) for | q | = 1 . Prop osition 7.11 . F or | q | = 1 the algebr a O q ( C × ⋉ C ) is an Ar ens-Michael algebr a, and the natur al inclusion R q ( C × ⋉ C ) ⊆ O q ( C × ⋉ C ) is an Ar ens-Michael envelop e of the algebr a R q ( C × ⋉ C ) : R q ( C × ⋉ C ) ♥ = O q ( C × ⋉ C ) Pr o of. This follows from the fact that for | q | = 1 the seminorms (7.61) o n R q ( C × ⋉ C ) are submultiplica- tive. R ( C × ) ⊙ z δ q R ( C ) ♥ → O ( C × ) ⊙ z δ q R ⋆ ( C ) for | q | 6 = 1 . Recall that the structure of algebras R ( C × ) and O ( C × ) w as discussed in § 3 (c). In particular, we hav e noted ther e that the topolo gy of O ( C × ) is generated by seminor ms (3.47): || u || C = X n ∈ Z | u n | · C | n | , C > 1 . In Ex ample 5.2 noted also that these seminorms are submultiplicativ e. Let us state tw o more their prop erties: ﬁr st, o b viously , C 6 D = ⇒ k u k C 6 k u k D (7.92) and, second, for | q | < 1 w e hav e | q | n 6 | q | −| n | , n ∈ Z , (7.93) This implies k M ⋆ δ q ( u ) k C = X n ∈ Z | u n | · | q | n · C | n | 6 X n ∈ Z | u n | · | q | −| n | · C | n | = X n ∈ Z | u n | ·  C | q |  | n | = k u k C / | q | i.e., ∀ i ∈ N   ( M ⋆ δ q ) i ( u )   C 6 k u k C / | q | i ( | q | < 1) (7 .94) Similarly , for | q | > 1 we have | q | n 6 | q | | n | , n ∈ Z , (7.9 5) so k M ⋆ δ q ( u ) k C = X n ∈ Z | u n | · | q | n · C | n | 6 X n ∈ Z | u n | · | q | | n | · C | n | = X n ∈ Z | u n | · ( C · | q | ) | n | = k u k C ·| q | i.e., ∀ i ∈ N   ( M ⋆ δ q ) i ( u )   C 6 k u k C ·| q | i ( | q | > 1) (7.96) In acc ordance with Pro positio n 7.10, let us identify in calcula tions the symbo ls 1 ⊡ t and z ⊡ 1 with the symbols z and t : 1 ⊡ t ≡ t, z ⊡ 1 ≡ z Lemma 7.8. If | q | 6 = 1 and r is a submu ltiplic ative seminorm on R ( C × ) z ⊙ δ q R ( C ) , then for some K ∈ N ∀ k > K r ( t k ) = 0 (7.97) § 7. APPE NDIX: HOLO MORPHIC REFLEXIVITY OF THE QUANTUM GR OUP ‘ AZ + B ’ 127 Pr o of. 1. Suppo se ﬁrst that | q | < 1 a nd take K ∈ N such that | q | K < 1 r ( z ) · r ( z − 1 ) . Then for any k > K we obtain | q | k · r ( z ) · r ( z − 1 ) < 1, and so r ( t k ) = r ( t k · z l · z − l ) = | q | lk · r ( z l · t k · z − l ) 6 | q | lk · r ( z l ) · r ( t k ) · r ( z − l ) 6 6 | q | lk · r ( z ) l · r ( t ) k · r ( z − 1 ) l = r ( t ) k · ( | q | k · r ( z ) · r ( z − 1 )) l − → l →∞ 0 2. On the contrary , supp ose | q | > 1. Then we ca n ta k e K ∈ N such that | q | K > r ( z ) · r ( z − 1 ), and for all k > K we hav e r ( z ) · r ( z − 1 ) | q | k < 1, so r ( t k ) = r ( t k · z − l · z l ) = | q | − lk · r ( z − l · t k · z l ) 6 | q | − lk · r ( z l ) · r ( t k ) · r ( z − l ) 6 6 | q | − lk · r ( z ) l · r ( t ) k · r ( z − 1 ) l = r ( t ) k ·  r ( z ) · r ( z − 1 ) | q | k  l − → l →∞ 0 Prop osition 7.12. 1) Supp ose | q | < 1 , and D > 1 and K ∈ N ar e such that D · | q | K > 1 (7.98) Then the seminorm p D,K : R ( C × ) z ⊙ δ q R ( C ) → R + , deﬁn e d by the e quality p D,K X k ∈ N u k ⊙ t k ! = K X k =0 || u k || D ·| q | k | {z } seminorm (3.47) = K X k =0 X n ∈ Z | u k,n | ·  D · | q | k  | n | (7.99) is s u bmultiplic ative. On t he c ontr ary, every su bmupltiplic ative seminorm on R ( C × ) z ⊙ δ q R ( C ) is sub or dinate to a seminorm of the form (7.99) . 2) If | q | > 1 , and D > 1 and K ∈ N ar e su ch that D | q | K > 1 (7.100) then the seminorm p D,K : R ( C × ) z ⊙ δ q R ( C ) → R + , deﬁn e d by the e quality p D,K X k ∈ N u k ⊙ t k ! = K X k =0 || u k || D | q | K | {z } seminorm (3.47) = K X k =0 X n ∈ Z | u k,n | ·  D | q | K  | n | (7.101) is submultiplic ative. On t he c ontr ary, every submultiplic ative seminorm on R ( C × ) z ⊙ δ q R ( C ) is sub or dinate to some seminorm of the form (7.99) . Pr o of. Consider the case | q | < 1. The submultiplicativit y of the seminor m (7.99) is veriﬁed dir ectly: p D,K ( u · v ) = K X k =0 k ( u · v ) k k D ·| q | k = (7.19) = K X k =0      k X i =0 u i · ( M ⋆ δ q ) i ( v k − i )      D ·| q | k = = K X k =0 k X i =0   u i · ( M ⋆ δ q ) i ( v k − i )   D ·| q | k 6 K X k =0 k X i =0 k u i k D ·| q | k ·   ( M ⋆ δ q ) i ( v k − i )   D ·| q | k 6 (7.92) , (7.94) 6 128 6 K X k =0 k X i =0 k u i k D ·| q | i · k v k − i k D ·| q | k − i 6 K X i =0 k u i k D ·| q | i ! ·   K X j =0 k v j k D ·| q | j   = p D,K ( u ) · p D,K ( v ) Then, let r b e a submultiplicative s eminorm o n R ( C × ) z ⊙ δ q R ( C ). Let us choose by Lemma 7.8 a n integer K ∈ N such that (7.97) holds , and put L = max 0 6 k 6 K r ( t k ) Note tha t r is a s ubm ultiplicative seminorm on the subalg ebra 1 ⊙ R ( C × ), consisting of functions o f the form a ⊙ 1, a ∈ R ( C × ), and isomo rphic to R ( C × ). Hence, on this subalg ebra r , must b e sub ordinate to some seminorm (3.47): r ( a ⊙ 1) 6 M · k a k C , a ∈ R ( C × ) (7.102) for some C > 1, M > 0 . Cho ose now D > 1 such that C 6 D · | q | K 6 D · | q | K − 1 6 ... 6 D · | q | 6 D (7.103) Then r ( u ) = r X k ∈ N u k ⊙ t k ! 6 X k ∈ N r ( u k ) · r ( t k ) = K X k =0 r ( u k ) · r ( t k ) 6 K X k =0 M · k u k k C · L = = L · M · K X k =0 k u k k D ·| q | k = L · M · p D,K ( u ) Thu s, r is sub ordinate to some seminorm p D,K . The case | q | > 1 is co nsidered simila rly , but instead of (7.94) we have to apply her e for m ula (7.96), and instead of (7.103) the chain C 6 D | q | K 6 D | q | K − 1 6 ... 6 D | q | 6 D Prop osition 7.13. F or | q | 6 = 1 the formula ζ n ⊙ t k 7→ z n ⊙ τ k (7.104) uniquely deﬁnes a homomorphism of Hopf algebr as R ( C × ) z ⊙ δ q R ( C ) → O ( C × ) z ⊙ δ q R ⋆ ( C ) , which is an Ar ens- Mich ael envelop e of the algebr a R ( C × ) z ⊙ δ q R ( C ) . Pr o of. It remains to chec k her e that O ( C × ) ⊙ R ⋆ ( C ) is a completion of R ( C × ) ⊙ R ( C ) with res pect to the seminorms (7.99) o r, dep ending on q , seminor ms (7.101). O ⋆ ( C × ) ⊙ δ q z R ( C ) ♥ → R ⋆ ( C × ) ⊙ δ q z R ⋆ ( C ) fo r arbitrary q . Consider the spa ce O ⋆ ( C × ). F rom the ﬁrst formula in (7.85) M ⋆ z ( ζ n ) = ζ n − 1 , one can deduce the ide n tit y: ( M ⋆ z ) k ( α ) = ( M ⋆ z ) k X n ∈ Z α n · ζ n ! = X n ∈ Z α n · ζ n − k = X m ∈ Z α m + k · ζ m (7.105) § 7. APPE NDIX: HOLO MORPHIC REFLEXIVITY OF THE QUANTUM GR OUP ‘ AZ + B ’ 129 Recall seminor ms || · || N on O ⋆ ( C × ), deﬁned by for m ulas (6 .12): k α k N = X | n | 6 N | α n | , N ∈ N , α = X n ∈ Z α n · ζ n Note the following their pr oper ties: M 6 N = ⇒ || β || M 6 || β || N (7.106) and   ( M ⋆ z ) i ( β )   M = X | n | 6 M | ( M ⋆ z ) i ( β ) n | = (7.105) = X | n | 6 M | β n + i | 6 X | n | 6 M + i | β n | = k β k M + i (7.107) By Theorem (7.6), the ge neral formula for multiplication in the algebra O ⋆ ( C × ) δ q ⊙ z R ( C ) has the form α · β = X k ∈ N k X i =0 α i ∗ ( M ⋆ z ) i ( β k − i ) ! ⊙ t k (7.108) and on basis elements lo oks as follows: ζ m ⊙ t k · ζ n ⊙ t l = (7.14) = ζ m ∗ ( M ⋆ z ) k ( ζ n ) ⊙ t k + l = (7.85) = = ζ m ∗ ζ n − k ⊙ t k + l = ( ζ m ⊙ t k + l , m = n − k 0 , m 6 = n − k = ( ζ m ⊙ t k + l , m + k = n 0 , m + k 6 = n (7.109) Lemma 7.9. Every c ontinuous su bmultiplic ative seminorm p on the algebr a O ⋆ ( C × ) δ q ⊙ z R ( C ) vanishes on almost al l element s of t he b asis { ζ n ⊙ t k ; n ∈ Z , k ∈ N } (i.e. on al l but a ﬁn ite subfamily): card { ( n, k ) ∈ Z × N : p  ζ n ⊙ t k  6 = 0 } < ∞ Pr o of. Put p n,k = p ( ζ n ⊙ t k ) and note that p m,k + l 6 p m,k · p m + k,l , k , m ∈ N , n ∈ Z (7.110) Indeed, from (7.1 09) it follows that ζ m ⊙ t k + l = ζ m ⊙ t k · ζ m + k ⊙ t l so p m,k + l = p ( ζ m ⊙ t k + l ) 6 p ( ζ m ⊙ t k ) · p ( ζ m + k ⊙ t l ) = p m,k · p m + k,l Consider the sets S k = { n ∈ Z : p n,k 6 = 0 } and note the following tw o things. 1. F rom the co n tinuit y a nd submupltip licativity of a seminorm p on R ( C ) δ q ⊙ z O ⋆ ( C × ) it follows that the functional p 0 ( α ) = p ( α ⊙ 1) = p ( α ⊙ t 0 ) , α ∈ O ⋆ ( C × ) is a contin uous submultiplicativ e s eminorm o n O ⋆ ( C × ): p 0 ( α ∗ β ) = p (( α ∗ β ) ⊙ 1) = p  ( α ⊙ 1) · ( β ⊙ 1)  6 p ( α ⊙ 1) · p ( β ⊙ 1) = p 0 ( α ) · p 0 ( β ) 130 This means, by Lemma 6 .2, that p 0 is sub ordinate to some seminorm of the fo rm (6.12): p 0 ( α ) 6 C · k α k N = C · X | n | 6 N | α n | , α ∈ O ⋆ ( C × ) This in its turn implies tha t the set S 0 = { n ∈ Z : p 0 ( ζ n ) = p (1 ⊙ ζ n ) 6 = 0 } must b e ﬁnite: card S 0 < ∞ 2. F rom the inequalities (7.110) one can deduce the following implications: ( p m,k +1 6 p m,k · p m + k, 1 p m,k +1 6 p m,k +1 · p m + k +1 , 0 = ⇒ ( S k +1 ⊆ S k S k +1 ⊆ S 0 − ( k + 1) = ⇒ S k +1 ⊆ S k ∩ ( S 0 − k − 1) (here S − i mea ns the shift of the set S by i units to left on the gr oup Z ). So we can conclude that S k form a tap ering chain of ﬁnite (since S 0 is ﬁnite) sets: S 0 ⊇ S 1 ⊇ ... ⊇ S k ⊇ S k +1 ⊇ ... And at the minimum after the num ber K = max S 0 − min S 0 this chain v anishes: S 0 ⊇ S 1 ⊇ ... ⊇ S K ⊇ S K +1 = ∅ (since S K +1 ⊆ S 0 ∩ ( S 0 − K − 1) = ∅ ). Prop osition 7.14. F or any N ∈ N the funct ional r N ( α ) = r N X k ∈ N α k ⊙ t k ! = N X k =0 k α k k N − k = N X k =0 X | n | 6 N − k | α n,k | (7.111) is a c ontinuous submultiplic ative seminorm on R ( C ) δ q ⊙ z O ⋆ ( C × ) . Every c ont inuous submu ltipli c ative semi- norm on R ( C ) δ q ⊙ z O ⋆ ( C × ) is sub or dinate d to some seminorm r N . Pr o of. The functional r N is a contin uous se minorm b ecause α 7→ α k are line ar co n tinuous mappings: r N ( λ · α + β ) = N X k =0 k λ · α k + β k k N − k 6 | λ | · N X k =0 k α k k N − k + N X k =0 k β k k N − k = | λ | · r N ( α ) + r N ( β ) Let us conside r the submultiplicativit y: r N ( α · β ) = (7.1 08) = r N X k ∈ N k X i =0 α i ∗ ( M ⋆ z ) i ( β k − i ) ⊙ t k ! = N X k =0      k X i =0 α i ∗ ( M ⋆ z ) i ( β k − i )      N − k 6 6 N X k =0 k X i =0 k α i k N − k ·   ( M ⋆ z ) i ( β k − i )   N − k 6 (7.107) 6 N X k =0 k X i =0 k α i k N − k · k β k − i k N − k + i 6 6 N X k =0 k X i =0 k α i k N − k | {z } 6 k α i k N − i , since k > i, hence N − k 6 N − i, and this allows to app ly (7. 106) · k β k − i k N − ( k − i ) 6 N X k =0 k X i =0 k α i k N − i · k β k − i k N − ( k − i ) 6 § 7. APPE NDIX: HOLO MORPHIC REFLEXIVITY OF THE QUANTUM GR OUP ‘ AZ + B ’ 131 6 N X i =0 k α i k N − i ! ·   N X j =0 k β j k N − j   = r N ( α ) · r N ( β ) It remains to c hec k that ev ery con tinuous subm ultiplicative seminorm p on R ( C ) δ q ⊙ z O ⋆ ( C × ) is subor dinate to some seminorm r N . This follows fr om Lemma 7.9: since p v a nishes on almost all basis elemen ts t k ⊙ ζ n , we can ﬁnd a num b er N ∈ N s uc h that { ( n, k ) ∈ Z × N : p ( ζ n ⊙ t k ) 6 = 0 } ⊆ { ( n, k ) ∈ Z × N : k 6 N & | n | 6 N − k } Then we put C = max { p ( ζ n ⊙ t k ); ( n, k ) : k 6 N , | n | 6 N − k } and obtain: p ( α ) = p   X n ∈ Z ,k ∈ N α n,k · ζ n ⊙ t k   6 X n ∈ Z ,k ∈ N | α n,k | · p ( ζ n ⊙ t k ) = N X k =0 X | n | 6 N − k | α n,k | · p ( ζ n ⊙ t k ) 6 6 C · N X k =0 X | n | 6 N − k | α n,k | = (7.111) = C · r N ( α ) Prop osition 7.15. 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A nal. i Prilozen. 8(1):75-76, 1974. [44℄ L. I. V ainerman, G. I. Ka. Non unimo dular ring groups, and Hopf-v on Neumann algebras. (Russian) Dokl. A kad. Nauk SSSR. 211: 1031-1034, 1973. [45℄ L. I. V ainerman, G. I. Ka. Non unimo dular ring groups and Hopf-v on Neumann algebras. (Russian) Mat. Sb. (N.S.) 94(136):194-225, 1974. [46℄ E. B. Vin b erg, A. L. Onish hik. Seminar p o gruppam Li i algebrai heskim gruppam. (Russian) [A seminar on Lie groups and algebrai groups℄ Seond edition. URSS, Moso w, 1995. 344 pp. ISBN: 5-88417-083-1 [47℄ S. W ang. Quan tum `ax+b' group as quan tum automorphism group of k [ x ] , h 70 94 v2 . [48℄ S. L. W orono wiz. Quan tum `az + b' group on omplex plane. Int. J. Math. 12(4):461-503, 2001. [49℄ S. Zhang, Braided Hopf algebras, arXiv:math.RA/0511251 v8 25 Ma y 2006. Con ten ts Int ro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 § 0 Ster eot yp e spa ces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 (a) Deﬁnition and typical exa mples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 (b) Smith s pace ge nerated by a co mpact set . . . . . . . . . . . . . . . . . . . . . . . . 8 (c) Brauner spaces genera ted by an expanding seq uence of co mpact sets . . . . . . . . 10 (d) Pro jective Banach sys tems and injective Smith sy stems . . . . . . . . . . . . . . . 10 (e) Banac h r epresent ation of a Smith space . . . . . . . . . . . . . . . . . . . . . . . . 12 (f ) Injective sy stems of B anach spaces , gener ated by compac t sets . . . . . . . . . . . 1 3 (g) Nuclear stereotype spa ces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6 (h) Spaces C M and C M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Space of functions C M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 The s pace o f p oint charges C M . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Dualit y b et ween C M and C M . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Bases in C M and C M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 § 1 Ster eot yp e Hopf a lgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 (a) T ensor pro ducts a nd the structure of monoidal catego ry on Ste . . . . . . . . . . . 21 (b) Stereotype Hopf algebra s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Algebras, coalgebr as a nd Hopf alg ebras in a sy mmetric monoida l categ ory . . 23 Pro jective and injective stereotype algebra s. . . . . . . . . . . . . . . . . . . 2 5 Stereotype Hopf algebr as. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Dualit y for stereo t yp e Hopf alg ebras. . . . . . . . . . . . . . . . . . . . . . . 26 Dual pa irs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (c) Key example: Hopf algebr as C G and C G . . . . . . . . . . . . . . . . . . . . . . . . 27 Algebra C G of functions on G . . . . . . . . . . . . . . . . . . . . . . . . . . 2 7 The a lgebra C G of p oint charges o n G . . . . . . . . . . . . . . . . . . . . . . 27 C G and C G as stereotype Hopf alg ebras. . . . . . . . . . . . . . . . . . . . . 2 8 (d) Sw eedler’s notations and the stereo t yp e appr o ximation prop erty . . . . . . . . . . 34 (e) Grouplik e elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 § 2 Stein ma nifolds: re ctangles in O ( M ) and rho m buses in O ⋆ ( M ) . . . . . . . . . . . . . . . 37 (a) Stein manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (b) Outer en velope s on M and r ectangles in O ( M ) . . . . . . . . . . . . . . . . . . . . 38 Op erations  and  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Outer env elop es on M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Rectangles in O ( M ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (c) Lemma on p olars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (d) Inner envelopes o n M and rho m buses in O ⋆ ( M ) . . . . . . . . . . . . . . . . . . . 43 Op erations  and ♦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Inner env elop es on M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Rhombuses in O ⋆ ( M ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 (e) Du ality b etw een rectangles and rhombuses . . . . . . . . . . . . . . . . . . . . . . 4 6 § 3 Stein g roups and Hopf algebra s co nnected to them . . . . . . . . . . . . . . . . . . . . . . 48 135 136 CONTENTS (a) Stein gr oups, linear gr oups and algebra ic groups . . . . . . . . . . . . . . . . . . . 4 8 (b) Hopf alg ebras O ( G ), O ⋆ ( G ), R ( G ), R ⋆ ( G ) . . . . . . . . . . . . . . . . . . . . . . 50 Hopf algebra s O ( G ) and O ⋆ ( G ) on a Stein gro up G . . . . . . . . . . . . . . 50 Hopf algebra s R ( G ) a nd R ⋆ ( G ) on an aﬃne algebra ic group G . . . . . . . . 51 Conv olutions in R ⋆ ( G ) and O ⋆ ( G ). . . . . . . . . . . . . . . . . . . . . . . . 51 (c) Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Algebras O ( Z ) and O ⋆ ( Z ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Algebras R ( C × ), R ⋆ ( C × ), O ( C × ), O ⋆ ( C × ). . . . . . . . . . . . . . . . . . . 53 The chain R ( C ) ⊂ O ( C ) ⊂ O ⋆ ( C ) ⊂ R ⋆ ( C ). . . . . . . . . . . . . . . . . . . 57 § 4 F unctions o f exp onential type on a Stein g roup . . . . . . . . . . . . . . . . . . . . . . . . 6 3 (a) Semicharacters and inverse semicharacters on Stein gro ups . . . . . . . . . . . . . 63 (b) Subm ultiplicative rhombuses a nd dually s ubm ultiplicative rectangles . . . . . . . . 66 (c) Holomorphic functions of exp onential type . . . . . . . . . . . . . . . . . . . . . . . 69 Algebra O exp ( G ) of holomor phic functions of exp onential type. . . . . . . . 69 Algebra O ⋆ exp ( G ) of exp onent ial analy tic functionals. . . . . . . . . . . . . . 71 (d) Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Finite groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Groups C n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Groups GL n ( C ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 (e) Injection ♭ G : O exp ( G ) → O ( G ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 (f ) Nuclea rit y o f the s paces O exp ( G ) and O ⋆ exp ( G ) . . . . . . . . . . . . . . . . . . . . . 76 (g) Holomor phic mappings of exp onential t yp e and tensor pro ducts of the spaces O exp ( G ) and O ⋆ exp ( G ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 (h) Structure o f Ho pf algebra s on O exp ( G ) and O ⋆ exp ( G ) . . . . . . . . . . . . . . . . . . 86 § 5 Are ns-Mic hael env elope a nd holomor phic reﬂexiv it y . . . . . . . . . . . . . . . . . . . . . 8 6 (a) Submultiplicativ e seminorms and Arens-Michael a lgebras . . . . . . . . . . . . . . 86 (b) Arens-Michael env elope s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 (c) The mapping ♭ ⋆ G : O ⋆ ( G ) → O ⋆ exp ( G ) is an Arens -Mic hael e n velope . . . . . . . . . 89 (d) The ma pping ♭ G : O exp ( G ) → O ( G ) is an Arens-Michael env elop e for the groups with algebra ic connected comp onent o f identit y . . . . . . . . . . . . . . . . . . . . 89 (e) Holomorphic reﬂexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 § 6 Holo morphic reﬂexiv it y as a generaliza tion of Pon tryagin duality . . . . . . . . . . . . . . 96 (a) Pont ryagin duality for compactly gener ated Stein gro ups . . . . . . . . . . . . . . . 96 (b) F our ier transfor m as an Arens- Mic hael envelope . . . . . . . . . . . . . . . . . . . . 96 Finite Abelia n group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 8 Complex plane C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Complex circle C × . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 0 Group of in teger s Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Pro of of Theorem 6 .2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 (c) Inclusion diagr am . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5 § 7 App endix: holomor phic reﬂexivity of the quantum group ‘ az + b ’ . . . . . . . . . . . . . . 106 (a) Quantum combinatorial formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 (b) Hopf alg ebra of skew po lynomials and s imilar co nstructions . . . . . . . . . . . . . 1 07 T ensor pro ducts X ⊙ R ( C ), X ⊛ R ( C ), X ⊙ R ⋆ ( C ), X ⊛ R ⋆ ( C ) . . . . . . . 107 Algebras of skew p olynomials A ϕ ⊙ R ( C ) and skew p ow er s eries A ϕ ⊛ R ⋆ ( C ) 1 08 Quantum pairs in stereotype Hopf a lgebras. . . . . . . . . . . . . . . . . . . 108 Hopf algebra s H ⊙ z ω R ( C ) and H ⋆ ⊛ ω z R ⋆ ( C ). . . . . . . . . . . . . . . . . . 109 Chains H ⊙ z ω R ( C ) ⊂ H ⊙ z ω O ( C ) ⊂ H ⊙ z ω O ⋆ ( C ) ⊂ H ⊙ z ω R ⋆ ( C ) and H ⊛ z ω R ( C ) ⊂ H ⊛ z ω O ( C ) ⊂ H ⊛ z ω O ⋆ ( C ) ⊂ H ⊛ z ω R ⋆ ( C ). . 115 (c) Quan tum g roup ‘ az + b ′ = R q ( C × ⋉ C ) . . . . . . . . . . . . . . . . . . . . . . . . 117 The g roup C × ⋉ C o f aﬃne tra nsformation of a complex plane. . . . . . . . 117 Stereotype algebr as R ( C × ⋉ C ) and R ⋆ ( C × ⋉ C ). . . . . . . . . . . . . . . . 11 7 CONTENTS 137 Stereotype algebr as O ( C × ⋉ C ) and O ⋆ ( C × ⋉ C ). . . . . . . . . . . . . . . . 120 R ( C × ⋉ C ), R ⋆ ( C × ⋉ C ), O ( C × ⋉ C ) and O ⋆ ( C × ⋉ C ) as Hopf a lgebras. . 120 Hopf algebra s R q ( C × ⋉ C ), R ⋆ q ( C × ⋉ C ), O q ( C × ⋉ C ), O ⋆ q ( C × ⋉ C ). . . . . 122 R q ( C × ⋉ C ) as a n a lgebra with genera tors and deﬁning r elations. . . . . . . 124 (d) Reﬂexivit y of R q ( C × ⋉ C ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 25 Reﬂexivity diagra m for R q ( C × ⋉ C ). . . . . . . . . . . . . . . . . . . . . . . 125 R q ( C × ⋉ C ) ♥ → O q ( C × ⋉ C ) for | q | = 1. . . . . . . . . . . . . . . . . . . . . 126 R ( C × ) ⊙ z δ q R ( C ) ♥ → O ( C × ) ⊙ z δ q R ⋆ ( C ) for | q | 6 = 1. . . . . . . . . . . . . . . . 126 O ⋆ ( C × ) ⊙ δ q z R ( C ) ♥ → R ⋆ ( C × ) ⊙ δ q z R ⋆ ( C ) for arbitrar y q . . . . . . . . . . . . 128 Bibliogra ph y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Conten ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Holomorphic Functions of Exponential Type and Duality for Stein Groups with Algebraic Connected Component of Identity

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