Unbounded bivariant $K$-theory and correspondences in noncommutative geometry

UNBOUNDED BIV ARIANT K -THEOR Y AND CORRESPONDE NCES IN NONCOMMUT A T IVE GEOMETR Y BRAM MESLAND Abstract. By int ro ducing a notion of smo oth connection for un bounded K K - cycles, we show that the Kaspa rov product of such cycles can b e deﬁned di - rectly , b y an algebraic formula. In order to ac hiev e this it is necessary to dev elop a framew ork of sm o oth algebras and a no tion of d iﬀerent iable C ∗ - module. The theory of operator spaces provides the required tools. Finally , the ab ov e men tioned K K -cycles with connect ion can b e viewe d as the mor- phisms in a category whose ob jects are sp ectral triples. Keyw ords: K K -theory; Kasparov product; sp ectral triples; op erator mo dules. Contents Int ro duction 2 Ac knowledgemen ts 5 1. C ∗ -mo dules 5 1.1. C ∗ -mo dules a nd their endomorphism algebra s 5 1.2. T ensor pr o ducts 6 1.3. Un bounded op era tors 7 2. K K -theory 11 2.1. The b ounded picture 11 2.2. The unbounded picture 12 3. Op erator modules 13 3.1. Oper ator spa ces 13 3.2. The Haa gerup tensor pro duct 14 3.3. Stably r igged mo dules 16 4. Smo othness 19 4.1. Sobo lev alg ebras 20 4.2. Holomorphic s tability 23 4.3. Smoo th C ∗ -mo dules 25 4.4. Inner pr o ducts, sta bilization a nd tensor pro ducts 27 4.5. Regular o p erators on C k -mo dules 30 4.6. T ransverse smo othness 33 4.7. Bounded p er turbations 34 5. Connections 37 5.1. Univ ersal fo rms 37 5.2. Pro duct co nnections 39 5.3. Smoo th co nnections 41 Date : Nov e mber 26, 2024. Key wor ds and phr ases. K K -theory , Kasparo v product, sp ectral triples, operator mo dules. 1 2 BRAM M ESLAND 5.4. Induced op er ators a nd their gra phs 42 5.5. Endomorphism algebras 44 6. Corresp o ndences 46 6.1. Almost a nticommut ing op er ators 47 6.2. The pr o duct of transverse mo dules 53 6.3. The K K - pr o duct 58 6.4. A categ ory of sp ectral triples 61 Appendix A. Smo othness and regular ity 62 Appendix B . Non unital C k -algebra s 64 References 66 Introduction Spec tr al tr iples [11] a re a central notion in Connes’ noncommutativ e geometry . The data for a spectral triple consist of a Z / 2 -gra ded C ∗ -algebra A , acting on a likewise graded Hilb ert space H , and a selfadjoint unbounded odd oper a tor D in H , with c ompact r esolven t, such that the suba lgebra A := { a ∈ A : [ D, a ] ∈ B ( H ) } , is dense in A . The ab ov e commutator is understo o d to b e graded. The motiv ating example is the Dirac op erator acting on the Hilbert space o f L 2 -sections of a com- pact spin manifold M . The C ∗ -algebra in question is then just C ( M ). Over the years, many noncommutativ e ex a mples of this structure hav e arise n, in par ticular in foliation theory [13] and examples dealing with non-pr op er gr oup actions. Shortly after Connes intro duction of sp ectral triples as cycles for K -homology [12], Baa j and Julg [2] generalized this notion to a biv ariant setting, b y r eplacing the H ilber t space H b y a C ∗ -mo dule E ov er a second C ∗ -algebra B . The notion of unbounded o p er ator with compact r e solven t extends to C ∗ -mo dules, and the commutator co nditio n is left unchanged. Such an ob ject ( E , D ) ca n b e thought of as a ﬁeld o f sp ectral triples parametrized b y B . Baa j and Julg show ed, moreover, that suc h o b jects can be taken as the cycles for Ka sparov’s K K -theory [20], and the externa l pro duct in K K -theor y simpliﬁes in this p icture. It is given by an algebraic formula. The main to pic of this pa pe r is the constr uction of a ca tegory Ψ o f unbounded K K -cycles, together with a functor Ψ → K K , i.e. co mpo sition of morphisms in Ψ corresp onds to the Kaspar ov pro duct in K K -theor y . In order to achieve this, a notion of smo othness for sp ectra l triples is introduced, and this notion is weak er than that o f r egularity [1 1] (also known in the litera ture as Q C ∞ ). It is ba sed on the fact that a s elfadjoint oper ator in a Hilbe r t space H is again selfadjoint viewed as an op erato r in its o wn graph. Thu s, it induces an inv erse s y stem of Hilb ert spaces · · · → G ( D n ) → G ( D n − 1 ) → · · · G ( D ) → H , BIV ARIANT K -THEOR Y AND CORRESPONDENCES 3 its Sob olev chai n . The algebra A mentioned ab ove ca n b e given a n oper ator spa ce top ology by rea liz ing it as matr ices through the r epresentation π D 1 : a 7→  a 0 [ D , a ] ( − 1) ∂ a a  ∈ B ( H ⊕ H ) . These matrices pr eserve the g r aph of D , a nd as s uch, one ca n commute them with D . This leads o ne to co nsider the *-a lgebra of elements for which the co mmut ators [ D , π D 1 ( a )] are b ounded in G ( D ). Pro ceeding inductiv ely , this leads to an inv erse system · · · → A k → A k − 1 → · · · A → A, acting o n the Sob olev chain o f D . These are involutive op er ator algebr as , meaning that the in v olution a 7→ a ∗ is completely bounded. Note that this inv olution is diﬀerent f rom that in the con taining C ∗ -algebra . The deﬁnit ion of k -smo othness now entails that the algebr a A k be dense in A . In that case, the alg ebras A k turn out to b e stable under holomorphic functional calculus in A . A C k - algebr a will be a C ∗ -algebra tog ether with a ﬁxed C k -sp ectral triple in the ab ove sens e. This mimic ks the deﬁnition o f a ma nifo ld as a top o lo gical space eq uipp ed with extra structure. Subsequently we study a class of smo oth mo dules for such algebr as. Giv en a C ∗ - mo dule E over a suﬃciently smo oth C ∗ -algebra B , the existence of an appr oximate unit which is well b ehaved with r esp ect to the top o logy on B k , allows for the resolution o f E by diﬀerentiable submodules · · · ⊂ E k ⊂ E k − 1 ⊂ · · · ⊂ E 1 ⊂ E . The notions of adjointable and un bounded r egular op era to rs mak e sense on such mo dules, and yield prop erties analoguous to those in C ∗ -mo dules. In particalar , the algebras End ∗ B k ( E k ) a nd K B k ( E k ) a re inv olutive oper ator a lgebras. A similar t yp e of mo dule has bee n studied extensively by Blecher ([5], [6]) and the theory developed here makes essential use of his r esults. The Haagerup tensor pro duct plays a crucial rˆ ole. It linearizes the multiplication in algebra s of op erators on Hilber t spaces. A s suc h, we base the deﬁnition o f Ω 1 ( B k ), the noncommutativ e diﬀerential for ms, on it a nd we conside r connections ∇ : E k → E k ˜ ⊗ B k Ω 1 ( B k ) , on the smo o th submo dules of E . W hen ( H , D ) is a sp ectral triple for B , such that B acts on the Sob o le v chain of D up to degre e k , we can form the op era to r 1 ⊗ ∇ D : ( e ⊗ f ) 7→ ( − 1) ∂ e ( e ⊗ D f + ∇ D ( e ) f ) . Its k - th Sob elev spa ce is isomorphic to E k ˜ ⊗ B k G ( T k ). The notion o f smo othnes s also allows us to dea l with sums of se lfadjoint operator s. When the mo dule E comes equipp ed with a s e lfadjoint r egular op erator S in E k and the connection is 1-smo oth with resp ect to S , then the oper ator S ⊗ 1 + 1 ⊗ ∇ D , is selfa djo int in E k ˜ ⊗ B k H . Moreov er, we show it has compact resolv ent whenever bo th D and S do so, and th us that this o p erator deﬁnes a sp ectral triple for A whenever ( E , S, ∇ ) is a suﬃcie ntly smo oth K K -cycle with connection. More genera lly , C k -cycle is a triple ( E k , S, ∇ ) which is a ( A k , B k )-bimo dule E k with unbounded r egular op er ator S such tha t ( S ± i ) − 1 ∈ K B k ( E k ) and a suﬃciently 4 BRAM M ESLAND smo oth connection ∇ : E k → E k ˜ ⊗ B k Ω 1 ( B k ). The isomor phis m c lasses of s uch cycles are denoted Ψ k 0 ( A, B ). W e show that suc h cycles can be co mpo sed b y the following alg ebraic formula: Theorem ( 6.2.7 ) . The c omp osition of C k cycles with c onne ction ( E k , S, ∇ ) ◦ ( F k , T , ∇ ′ ) = ( E k ˜ ⊗ B k F k , S ⊗ 1 + 1 ⊗ ∇ T , 1 ⊗ ∇ ∇ ′ ) , yields a C k -cycle with c onne ction, a nd is asso ciative up to isomorphism. That is, this comp osition pr eserves al l smo othness pr op erties . Note that this pro duct is deﬁned on the level of the inv olutive op erator algebras A k coming fr o m the s p ectr al triple on A , a nd that the A k are not C ∗ -algebra s. Smo oth bimo dules can then b e interpreted as morphisms of sp ectr al triples. This can b e captured in a diagra m: A → ( H , D ) ⇌ C ↓ k ( E k , S, ∇ ) C ⇃↾ k B → ( H ′ , D ′ ) ⇌ C . W e us e the notation E ⇌ B to indicate that E , the C ∗ -completion of E k , is a C ∗ -mo dule ov er B . This also emphasizes the asymmtery , and hence the direction, of the morphisms. It seems appro priate to refer to a bimodule with connectio n ( E k , S, ∇ ) a s a ge ometric c orr esp ondenc e . The comp os ition of g eometric co rresp ondences is the un b ounded v er sion of the Kaspar ov pr o duct in K K -theo ry . Reca ll that the Kaspa rov pro duct ([2 0]) K K i ( A, B ) ⊗ K K j ( B , C ) → K K i + j ( A, C ) , allows one to view the K K -gro ups as mo r phisms in a ca tegory w ho se o b jects ar e all C ∗ -algebra s. K K is a triangulated category and is universal for C ∗ -stable, split- exact functors on the ca tegory of C ∗ -algebra s [18]. The degree of a K K - cycle is determined by the action of a Cliﬀor d algebra . In pa r ticular sp ectral triple s can be ass igned a degr ee. Denote the set of unitary is o morphism classes of k -smo oth geometric corr esp ondences of the above sp ectral tr iples , which we a ssume to ha ve degrees i and j , resp ectively , by Cor k ( D , D ′ ). The main result of this paper sta tes that Theorem ( 6 .4.2 ) . The b ounde d tr ansform b : D 7→ D (1 + D 2 ) − 1 2 deﬁnes a functor b : Co r k ( D , D ′ ) → K K i − j ( A, B ) ( E k , S, ∇ ) 7→ [( E , b ( D ))] . This is done by taking C ∗ -completions, and fo rgetting all smo othness and the connection. In particular it follows that the map K j ( B ) → K i ( A ) deﬁned b y the corres p o ndence maps the K - homology class of ( B , H ′ , D ′ ) to that of ( A, H , D ). BIV ARIANT K -THEOR Y AND CORR ESPONDENCES 5 The structur e of the pap er is as follows. In the ﬁrst three s e ctions we review the theor y of C ∗ -mo dules, unbounded op erator s, K K -theor y and op er ator mo dules. Some of this material is well known, but w e intro duce several constr uctions that will be used extensively later in the paper . W e describ e some res ults that ar e not stated explicitly in the literatur e , or emphasize the in terco nnection of t he theories. This should make the second part of the pap er an ea sier rea d. In section 4 we introduce smo othness for sp ectr a l triples and describ e the pr op erties of smo oth algebra s, smo oth mo dules , and op era to rs t hereon. F or theore tica l purposes this notion is easier to w ork with and it a llows for the deﬁnition of a g e neral notion of smo o th C ∗ -mo dule. In section 5 we adapt the theor y of connections to the op erator e mo dule setting and obtain results on the structure o f th e graphs of unbounded op era tors t wisted b y suc h a connec tio n. This is used in section 6 to show that the twisting construction is in fact the Kaspar ov product in dis guise. That in turn leads to the deﬁnition of the category o f sp ectral triples describ ed a b ov e. Ac knowledgemen ts . This pap er was conceived during my P h.D. studies at the Max P lanck Institut f¨ ur Mathematik in Bonn, Germany . The suppor t of Matilde Marcolli during this per io d has b een o f great v alue. The work was ﬁnalized during m y stay at Utre cht Universit y , the Netherlands. I thank both intitutions for t heir suppo rt. I am gra teful to Florida Sta te Univ er sity and the California Institut e of T e chnology for their hospitality and s uppo rt. Man y thanks a s w ell to Nigel Higson, for useful and motiv a ting corresp ondence and conv ersations. I thank Saa d Baa j, Alain Connes, Andre Henriqu´ es, Matthias Lesch, Uuye Otg o nba yer a nd W alter v an Suijlek om for useful corresp o ndence and discussions . I am indebted to Nikola y Iv anko v for carefully reading the man uscr ipt and numerous useful con v ersations. Finally I thank Javier Lop ez for several conv ersa tions we had in the early stages of this pro ject. 1. C ∗ -modules F r om the Gelfand-Naimar k theo rem w e know that C ∗ -algebra s are a natura l g en- eralization of loc ally c o mpact Hausdor ﬀ topolog ical space s . In the same vein, the Serre-Swan theore m tells us that ﬁnite pr o jective mo dules are analogues of lo cally trivial ﬁnite-dimensional complex vector bundles ov er a top ologica l space.T he sub- sequent theory of C ∗ -mo dules, pioneered by Pasc hke and Rieﬀel, should b e viewed in the ligh t o f these theorems. They ar e like Her mitian vector bundles ov er a space. 1.1. C ∗ -mo dul es and their endomorphism algebras. In the subsequent review of the esta blished theory , we will as sume all C ∗ -algebra s and H ilber t spa c e s to b e separable, and all mo dules to b e countably genera ted. This last as s umption means that there exis ts a countable set o f g enerator s whose alg ebraic span is dense in the mo dule. Deﬁnition 1.1.1. Let B b e a C ∗ -algebra . A right C ∗ - B - mo dule is a complex vector spa ce E which is also a right B - mo dule, equipp ed with a bilinea r pa iring E × E → B ( e 1 , e 2 ) 7→ h e 1 , e 2 i , such that • h e 1 , e 2 i = h e 2 , e 1 i ∗ , • h e 1 , e 2 b i = h e 1 , e 2 i b, 6 BRAM M ESLAND • h e, e i ≥ 0 and h e, e i = 0 ⇔ e = 0 , • E is c o mplete in the norm k e k 2 := k h e , e ik . W e use Landsman’s notatio n ([24]) E ⇌ B to indicate this structure. The closure of the linear span of elements of the for m h e 1 , e 2 i is an ideal in E .The mo dule E is said to b e ful l if this ide a l is all of B . F o r t wo such modules, E a nd F , one can consider opera tors T : E → F . As opp osed to the case of a Hilb ert s pace ( B = C ), such operator s need not alwa ys hav e an adjoint with resp ect to the inner pro duct. Therefore let Hom ∗ B ( E , F ) := { T : E → F : ∃ T ∗ : F → E , h T e 1 , e 2 i = h e 1 , T ∗ e 2 i} . Elements of Hom ∗ B ( E , F ) are called adjointable op er ators . When E = F , End ∗ B ( E ) denote the a djo intable endomorphisms of the C ∗ -mo dule E . It is a C ∗ -algebra a nd contains the canonical C ∗ -subalgebra of B - c omp act op er ators denoted by K B ( E ), constructed as fo llows. The inv olutio n on B allows for co nsidering E as a left B - mo dule via be := eb ∗ . The inner pr o duct can b e used to turn the algebraic tensor pro duct E ⊗ B E into a ∗ -alg ebra: e 1 ⊗ e 2 ◦ f 1 ⊗ f 2 := e 1 h e 2 , f 1 i ⊗ f 2 , ( e 1 ⊗ e 2 ) ∗ := e 2 ⊗ e 1 . This alg ebra is denoted by Fin B ( E ), and K B ( E ) is its norm closure. A gr ading on a C ∗ -algebra B is an elemen t ˆ γ ∈ Aut ∗ B (a *-automo rphism), of order 2 . If suc h a gr ading is presen t, B decompo s es a s B 0 ⊕ B 1 , where B 0 is the C ∗ -subalgebra of even elements, and B 1 the close d subspa c e of o dd elements. W e hav e B i B j ⊂ B i + j for i, j ∈ Z / 2 Z . F o r b ∈ B i , we denote the de gr e e of b by ∂ b ∈ Z / 2 Z . A gr ade d *-homomo rphism φ : A → B betw e en g raded C ∗ -algebra s, is a *- homomorphism that resp ects the gr adings, i.e. φ ◦ ˆ γ A = ˆ γ B ◦ φ . F rom now on, we as s ume all C ∗ -algebra s to b e gra ded, p ossibly trivially , i.e. ˆ γ = 1. Deﬁnition 1. 1.2. A C ∗ -mo dule E ⇌ B is gr ade d if it comes equipped with an element γ ∈ Aut C ( E ), of order 2, such that • γ ( eb ) = γ ( e ) ˆ γ ( b ) , • h γ ( e 1 ) , γ ( e 2 ) i = ˆ γ h e 1 , e 2 i . In this case E also decomp oses as E 0 ⊕ E 1 , and we hav e E i B j ⊂ E i + j for i, j ∈ Z / 2 Z . The a lg ebras End B ( E ) , End ∗ B ( E ) and K B ( E ) inherit a natura l g rading from E b y setting (ˆ γ T )( e ) := γ ( T γ ( e )). F or e ∈ E i , we denote the de gr e e of e b y ∂ e ∈ Z / 2 Z .F rom now on we assume all C ∗ -mo dules to b e gr aded, p ossibly tr ivially . 1.2. T ensor pro du cts. F or a pair of C ∗ -mo dules E ⇌ A and F ⇌ B , the vector space tens o r pr o duct E ⊗ F can b e made into a C ∗ -mo dule ov er the minimal C ∗ - tensor pro duct A ⊗ B . The minimal or sp atial C ∗ -tensor pr o duct is obtained as the closure of A ⊗ B in B ( H ⊗ K ), where H and K are gr a ded Hilbert spa ces that carry f aithful graded r epresentations of A and B r esp ectively . In order to mak e A ⊗ B in to a graded alg ebra, the mult iplication law is deﬁned as (1.1) ( a 1 ⊗ b 1 )( a 2 ⊗ b 2 ) = ( − 1 ) ∂ b 1 ∂ a 2 a 1 a 2 ⊗ b 1 b 2 . The completion of E ⊗ F in the inner pro duct h e 1 ⊗ f 1 , e 2 ⊗ f 2 i := h e 1 , e 2 i ⊗ h f 1 , f 2 i , is a C ∗ -mo dule denoted by E ⊗ F . It inherits a grading by setting γ := γ E ⊗ γ F . BIV ARIANT K -THEOR Y AND CORRESPONDENCES 7 The gr aded mo dule so obtained is the ext erior tensor pr o duct of E and F . The gr ade d tensor pr o duct of maps φ ∈ End ∗ A ( E ) and ψ ∈ End ∗ B ( F ) is deﬁned b y φ ⊗ ψ ( e ⊗ f ) := ( − 1) ∂ ( e ) ∂ ( ψ ) φ ( e ) ⊗ ψ ( f ) , gives a graded inclusion End ∗ A ( E ) ⊗ End ∗ B ( F ) → End ∗ A ⊗ B ( E ⊗ F ) , which res tricts to an isomorphism K A ( E ) ⊗ K B ( F ) → K A ⊗ B ( E ⊗ F ) . A * -homomorphis m A → End ∗ B ( E ) is said to be essential if A E := { n X i =0 a i e i : a i ∈ A, e i ∈ E , n ∈ N } , is dense in E . If a gra ded esse n tial *-homomor phis m B → E nd ∗ C ( F ) is giv en, one can complete the alg ebraic tensor pr o duct E ⊗ B F to a C ∗ -mo dule E ˜ ⊗ B F over C . The norm in which to co mplete c omes fro m the B -v a lued inner pro duct (1.2) h e 1 ⊗ f 1 , e 2 ⊗ f 2 i := h f 1 , h e 1 , e 2 i f 2 i . There is a * -homomor phis m End ∗ B ( E ) → End ∗ C ( E ˜ ⊗ B F ) T 7→ T ⊗ 1 , which res tricts to a homomorphism K B ( E ) → K C ( E ˜ ⊗ B F ).If E car ries an (essen- tial) A -repr esentation, then so do es E ˜ ⊗ B F . W e write H B for the graded tensor product H ˜ ⊗ C B , where H = ℓ 2 ( Z \ { 0 } ) ∼ = ℓ 2 ( N ) ⊕ ℓ 2 ( N ) with its usual gr ading. F or nonunital B o ne sets H B := H B + B . H B absorbs any countably gener a ted C ∗ -mo dule. The dir ect sum E ⊕ F of C ∗ - B - mo dules b ecomes a C ∗ -mo dule in the inner pro duct h ( e 1 , f 1 ) , ( e 2 , f 2 ) i := h e 1 , e 2 i + h f 1 , f 2 i . Theorem 1.2.1 (Kaspar ov [20]) . L et E ⇌ B b e a c ountably gener ate d gr ade d C ∗ -mo dule. Then ther e exists a gr ade d unitary isomorphism E ⊕ H B ∼ − → H B . 1.3. Unbounde d op erators. Similar to the Hilber t space setting, ther e is a notion of unbounded op era tor on a C ∗ -mo dule. Man y o f the alr e a dy subtle issues in the t heory of unbounded op erator s should be handled with even more care. This is mostly due to the fact that clo sed submo dules of a C ∗ -mo dule need not be orthogo nally co mplemen ted. W e refer to [1], [2 3] a nd [28] for detailed e xp o sitions of this theory . Deﬁnition 1.3. 1 ([2]) . Let E b e a C ∗ - B -mo dule. A densely deﬁned closed op er- ator D : Dom D → E is called r e gular if • D ∗ is densely deﬁned in E • 1 + D ∗ D has dense r ange. Such an op erato r is automa tically B -linear, and Do m D is a B -submo dule of E . There are tw o opera tors, r ( D ) , b ( D ) ∈ End ∗ B ( E ) c a nonically asso ciated with a regular op era tor D . They are the inverse mo dulus of D (1.3) r ( D ) := (1 + D ∗ D ) − 1 2 , 8 BRAM M ESLAND and the b ounde d tr ansform (1.4) b ( D ) := D (1 + D ∗ D ) − 1 2 . A regula r o pe rator D is symmet ric if Dom D ⊂ Dom D ∗ and D = D ∗ on Dom D . It is selfadjoint if it is symmetric and Dom D = Dom D ∗ . Prop ositi o n 1.3.2 . If D : Dom D → E is r e gu lar, then D ∗ D is selfadj oint and r e gular. Mor e over, Do m D ∗ D is a c or e for D and Imr ( D ) = Dom D . It follows tha t D is completely determined by b ( D ), as r ( D ) 2 = 1 − b ( D ) ∗ b ( D ). Recall that a submo dule F ⊂ E is c omplemente d if E ∼ = F ⊕ F ⊥ , w her e F ⊥ := { e ∈ E : ∀ f ∈ F h e, f i = 0 } . Contrary to the Hilbert space cas e , clos e d submo dules of a C ∗ -mo dule need not be complemented. The complemented submo dules of a C ∗ -mo dule E are precis e ly those o f the for m p E , with p a pr o jection in End ∗ B ( E ). The gr aph of D is the closed submo dule G ( D ) := { ( e, D e ) : e ∈ Dom ( D ) } ⊂ E ⊕ E . There is a canonical unitary v ∈ E nd ∗ B ( E ⊕ E ), deﬁned b y v ( e, f ) := ( − f , e ). Note that G ( D ) and v G ( D ∗ ) are or thogonal submo dules o f E ⊕ E . The following algebraic characterizatio n of regula rity is due to W or onowicz. Theorem 1.3. 3 ([28]) . A densely deﬁne d close d op er ator D : E → E , with densely deﬁne d adjoint is re gular if a nd o nly if G ( D ) ⊕ v G ( D ∗ ) ∼ = E ⊕ E . The isomorphism is given by co ordinatewise addition. Mo reov er, the op era tor (1.5) p D :=  r ( D ) 2 r ( D ) b ( D ) ∗ b ( D ) r ( D ) b ( D ) b ( D ) ∗  satisﬁes p 2 D = p ∗ D = p D , i.e. it is a pro jection, and p D ( E ⊕ E ) = G ( D ) . When D is an o dd operator , the gr ading γ ⊕ ( − γ ) on E ⊕ E resp ects the decompositon from theor em 1.3.3. W e will alwa ys consider E ⊕ E with this grading. In cas e D is selfadjoint, the ab ov e pro jection takes the form p D =  (1 + D 2 ) − 1 D (1 + D 2 ) − 1 D (1 + D 2 ) − 1 D 2 (1 + D 2 ) − 1  , so the comp onents a re a lgebraic functions of D . Moreover v pv ∗ = 1 − p in this c a se. These tw o facts will play a crucial rˆ ole in this pa per . The module G ( D ), which is naturally in bijection with Do m ( D ), inherits the structure of a C ∗ -mo dule fro m E ⊕ E . W e denote its inner pro duct b y h· , ·i 1 . It is a well known fact that r ( D ) 2 = ( D + i ) − 1 ( D − i ) − 1 , and the opera to rs D ± i are bijections Dom D → E . W e refer to the oper ators ( D ± i ) − 1 as the r esolvents o f D . Since D commutes with ( D ± i ) − 1 , D maps ( D ± i ) − 1 G ( D ) into G ( D ). W e denote this op er ator by D 2 . Prop ositi o n 1.3.4 . L et D : Do m D → E b e a selfdajoint re gular op er ator. Then D 2 : ( D ± i ) − 1 G ( D ) → G ( D ) is a selfad joint r e gular o p er ator. When D is o dd, so is D 2 . BIV ARIANT K -THEOR Y AND CORRESPONDENCES 9 Pr o of. F rom prop osition 1 .3.2 it follows that ( D ± i ) − 1 G ( D ) = r ( D ) 2 E = Dom D 2 . D 2 is closed as an op er ator in G ( D ) for if r ( D ) 2 e n → r ( D ) 2 e a nd D r ( D ) 2 e n → e ′ in the top ology of G ( D ), then it follows immedia tely that e ′ = D ( D r ( D ) 2 e ) = D 2 r ( D ) 2 e. It is straightforw ard to c hec k that D 2 is symmetric for the inner pr o duct of G ( D ). Hence it is regular, because (1 + D 2 ) r ( D ) 4 E = r ( D ) 2 E . T o pro ve selfadjoint ness, suppo se y ∈ Dom D is s uch tha t there exists z ∈ Dom D such that for all x ∈ r ( D ) 2 E h D 2 x, y i 1 = h x, z i 1 . Then z = D y , beca us e h D x, y i 1 = h Dx, y i + h D 2 x, Dy i = h D r ( D ) 2 e, y i + h D 2 r ( D ) 2 e, D y i = h r ( D ) 2 e, D y i + h D 2 r ( D ) 2 e, D y i = h e, D y i . A similar computation shows that h x, z i 1 = h e, z i . Since r ( D ) 2 is injective this holds for all e ∈ E , and hence z = D y . Therefore Dom D ∗ 2 = { y ∈ Dom D : D y ∈ Dom D } = Dom D 2 = r ( D ) 2 E = Dom D 2 , so D 2 is selfadjoint.  Corollary 1.3.5. A selfa djoint r e gular op er ator D : Dom D → E induc es a mor- phism of inverse systems of C ∗ -mo dules: · · · ✲ E i +1 ✲ E i ✲ E i − 1 ✲ · · · ✲ E 1 ✲ E · · · ✲ E i +1 ✲ D i +1 ✲ E i ✲ D i ✲ E i − 1 ✲ D i − 1 ✲ · · · ✲ D i − 2 ✲ E 1 ✲ D 2 ✲ E D 1 = D ✲ Pr o of. Set E i = G ( D i ) . Then t he maps E i → E i − 1 are just pro jection o n the ﬁrs t co ordinate, wher eas the maps D i : E i → E i − 1 are the pro jections on the second co ordinates. These maps ar e adjointable, and we have D ∗ i ( e i ) = ( D i r ( D i ) 2 e i , D 2 i r ( D i ) 2 e i ) , φ ∗ i ( e i ) = ( r ( D i ) 2 , D i r ( D i ) 2 ) . These ar e exa ctly the compo nents of the W oronowicz pro jection 1 .5.  W e will r efer to this inv erse system a s the Sob olev chain of D . Almost self- adjoint oper ators were int ro duced by Kucerovsky in [22]. They a r e adjointable per turbations of selfadjoint op erator s. Deﬁnition 1.3.6. Let D b e a regular o p er ator in a C ∗ - B -mo dule E . D is almost selfadjo int if Dom D = Dom D ∗ and D − D ∗ extends to an elemen t in End ∗ B ( E ) . The following result is implicit in [22]. Prop ositi o n 1.3 .7. L et D b e an almost selfadjoint op er ator on a C ∗ - B -mo dule E and b = D ∗ − D ∈ End ∗ B ( E ) . F or | λ | > k b k , t he op er ators D + λi , D ∗ − λi ar e bije ctions Dom D → E . 10 BRAM M ESLAND Pr o of. The o p er ator T := D + D ∗ is selfadjoint, and D = T + b . The op erator s T + λi are bijections Dom D → E , and k ( T + λi ) − 1 k ≤ 1 λ . Since ( D + λi )( T + λi ) − 1 = 1 + b ( T + λi ) − 1 , and 1 + b ( T + λi ) − 1 is inv ertible whenever | λ | > k b k , we see that D + λi is s urjective. It is injective b ecause h ( D + λi ) e, ( D + λi ) e i = h De , D e i − λi h e, D e i + λi h D e , e i + λ 2 h e, e i = h De , D e i − λi h be, e i + λ 2 h e, e i ≥ h ( λib + λ 2 ) e, e i + λ 2 h e, e i ≥ λ 2 h e, e i . Reversing the role s of D and D ∗ shows that D ∗ − λi is bijective as well.  Corollary 1.3. 8. L et D b e an almost selfadjoint r e gular op er ator in E . Then Dom D ∗ D = Dom D 2 , and the op er ator 1 + D 2 λ 2 : Dom D 2 → E , is bije ctive f or λ suﬃciently lar ge. Mor e over, deﬁne p := (1 + D 2 λ 2 ) − 1 D λ 2 (1 + D 2 λ 2 ) − 1 D (1 + D 2 λ 2 ) − 1 D 2 λ 2 (1 + D 2 λ 2 ) − 1 ! , v λ :=  0 − λ − 1 λ 0  , then p is an ide mp otent a nd v λ an invertible in End ∗ B ( E ) such that Im p = G ( D ) and v λ pv − 1 λ = 1 − p . Pr o of. Since Dom D ∗ = Do m D , w e hav e Dom D ∗ D = Dom D 2 . By prop osition 1 .3 .7 D ± λi a re bijections Dom D → E . Thus, λ 2 + D 2 = ( D + λi )( D − λi ) : Dom D 2 → E , bijectiv ely as well. Mo reov er, the in verse ( λ 2 + D 2 ) − 1 = ( D + λi ) − 1 ( D − λi ) − 1 is bo unded and a djointable. That p is idempotent is now easily chec ked, as w ell the prop erty v λ pv − 1 λ = 1 − p . It is immediate tha t Im p ⊂ G ( D ) and Im p ∗ ⊂ G ( D ∗ ). Therefore ker p = Im (1 − p ∗ ) = Im v p ∗ v ∗ ⊂ v G ( D ∗ ) , which implies that Im p = G ( D ).  Thu s, for a n almost selfadjoint op e rator there is an inv er tible adjointable op er- ator g : G ( D ) ⊕ v λ G ( D ) ∼ − → E ⊕ E , which is a k ey exa mple of the following deﬁnition. Deﬁnition 1.3.9. T wo C ∗ -mo dules E ⇌ B a nd F ⇌ B are top olo gic al ly isomor- phic if there are g ∈ Hom ∗ B ( E , F ) a nd g − 1 ∈ Ho m ∗ B ( F , E ) w ith g g − 1 = 1 F , g − 1 g = 1 E . Such a g is c a lled a top olo gic al isomorphi sm . Prop ositi o n 1.3.10. An almost selfadjoint op er ator D i s almost selfadjoi nt in its own gr aph, a nd henc e ind uc es a S ob olev chain as in the selfadjoint c ase. BIV ARIANT K -THEOR Y AND CORRESPONDENCES 11 Pr o of. W e deﬁne D in its own gra ph on the doma in ( D + λi ) − 1 G ( D ). This is dens e since Do m D ∗ D = Do m D 2 is a c o re for D . It is straightforward to chec k that D 2 is closed o n this domain, and that 2 R = D 2 − D ∗ 2 is b ounded adjointable. Moreov er, by deﬁnition D 2 + λi is s ur jective a nd has a djoint able inv erse. Therefore 1 2 ( D 2 + D ∗ 2 + λi )( D 2 + λi ) − 1 = 1 + R ( D 2 + λi ) − 1 , is inv ertible for λ s uﬃcie ntly la rge, and D 2 + D ∗ 2 is selfadjoint.  2. K K -theor y Kaspar ov’s biv a riant K -theory K K [20] has become a central to ol in noncom- m utative geometry since its crea tion. It is a bifunctor o n pair s of C ∗ -algebra s, asso ciating to ( A, B ) a Z / 2 Z -graded g roup K K ∗ ( A, B ). It uniﬁes K -theory and K -ho mology in the sense that K K ∗ ( C , B ) ∼ = K ∗ ( B ) and K K ∗ ( A, C ) ∼ = K ∗ ( A ) . Much of its usefulness comes from the existence of in terna l and external pro duct structures, b y which K K -elements induce homomorphisms b etw een K -theor y and K -ho mology gro ups. In Ka sparov’s original a pproach, the deﬁnition a nd computa- tion of the pro ducts is very complicated. In order to s implify the external pro duct, Baa j and Julg [2] introduced another mo del for K K , in which the external pro duct is giv e n by a simple algebraic formula. The price one ha s to pay is working with un bo unded oper ators. 2.1. The b ounded picture. The main idea behind Kasparov’s appro a ch to K - homology and K K -theory is that of a fa mily of abstra c t elliptic op erator s. This was an idea pioneer ed by Atiy ah, in his constructio n of K -homology for space s and the family index theorem. Deﬁnition 2.1. 1 ([20]) . Let A → E ⇌ B b e a graded bimo dule and F ∈ End ∗ B ( E ) an o dd op era tor. ( E , F ) is a Kasp ar ov ( A, B ) -bimo dule if, for all a ∈ A , • [ F, a ] , a ( F 2 − 1) , a ( F − F ∗ ) ∈ K B ( E ). The set o f Kas pa rov mo dules up to unitary equiv alence is denoted E 0 ( A, B ), and E j ( A, B ) := E 0 ( A, B ⊗ C j ), where C j is the j -th complex Cliﬀord algebr a. The set of de gener ate element s cons ists of bimodules for which ∀ a ∈ A : [ F, a ] = a ( F 2 − 1) = a ( F − F ∗ ) = 0 . Denote by e i : C [0 , 1] ⊗ B → B the e v a lution map at i ∈ [0 , 1]. Two K asparov ( A, B )-bimo dules ( E i , F i ) ∈ E j ( A, B ), i = 0 , 1 ar e homotopic if there exists a Kaspar ov ( A, C [0 , 1] ⊗ B )-mo dule ( E , F ) ∈ E j ( A, C [0 , 1] ⊗ B ) for which ( E ⊗ e i B , F ⊗ 1) is unitar ily equiv alent to ( E i , F i ), i = 0 , 1. It is an equiv alence relation, denoted ∼ . Deﬁne K K j ( A, B ) := E j ( A, B ) / ∼ . K K j is a bifunctor, contra v a riant in A , co v a riant in B , taking v a lues in ab e lian groups. It is not hard to sho w that K K ∗ ( C , A ) and K K ∗ ( A, C ) are na turally iso- morphic to the K -theor y and K -homolog y of A , resp ectively . Moreover, Kasparov prov ed the following deep theorem. 12 BRAM M ESLAND Theorem 2.1.2 ([20]) . F or any C ∗ -algebr as A, B , C ther e exists an asso ciative biline ar p airing K K i ( A, B ) ⊗ Z K K j ( B , C ) ⊗ B − − → K K i + j ( A, C ) . Ther efor e, the gr oups K K ∗ ( A, B ) ar e the morphism sets of a c ate gory K K whose obje cts ar e all C ∗ -algebr as. There a lso is a no tion of external pro duct in K K -theory . Theorem 2 . 1.3 ([20 ]) . F or any C ∗ -algebr as A, B , C, D ther e exists an asso ciative biline ar p airing K K i ( A, C ) ⊗ Z K K j ( B , D ) ⊗ − → K K i + j ( A ⊗ B , C ⊗ D ) . The external pr o duct makes K K into a symmetric m onoidal c ate gory The category K K ha s mor e re ma rk able pro pe rties. Although w e will not use them in this pap er, we do b elieve they deserve a brief men tion. It was shown b y Cun tz a nd Higs o n ([1 4],[18]) tha t the ca tegory K K is universal in the s e nse tha t any split exact stable functor from the catego ry of C ∗ -algebra s to, say , tha t of ab elian groups, factors through the category K K . Altoug h it fa ils to be abelian, K K is a triangulated category . This allows fo r the developmen t of homological algebra in it, which has sp ecial interest in relation to the Baum-Connes conjecture, a n approach pursued by Nest and Meyer [26]. 2.2. The unbounded picture. One can deﬁne K K -theory using un bo unded op- erators on C ∗ -mo dules. As th e b ounded deﬁnition corr esp onds to abstract order zero elliptic pseudo diﬀerential op era tors, the unbounded version cor resp onds to order o ne op erator s. Deﬁnition 2. 2.1 ([2 ]) . L et A → E ⇌ B be a gra de d bimo dule a nd D : Dom D → E an o dd selfadjoint regula r op erato r. The pair ( E , D ) is an K K -cycle for ( A, B ) if, for all a ∈ A , a dense subalge br a of A • a Dom D ⊂ Dom D and [ D , a ] ex tends to an ope r ator in End ∗ B ( E ) • a r ( D ) ∈ K B ( E ). Denote the set of KK -cycles for ( A, B ˜ ⊗ C i ) mo dulo unitary equiv a le nce b y Ψ i ( A, B ). As in the b ounded case , we will refer to elements of Ψ 0 as even unbounded bimo d- ules. In [2] it is shown that ( E , b ( D )) is a Ka sparov bimo dule, a nd that every element in K K ∗ ( A, B ) can b e repr e sented by an unbounded bimodule. The moti- v a tion for introducing un bo unded mo dules is the following r esult. Theorem 2. 2.2 ([2]) . L et ( E i , D i ) b e unb oun de d bimo dules for ( A i , B i ) , i = 1 , 2 . The op er ator D 1 ⊗ 1 + 1 ⊗ D 2 : Dom D 1 ⊗ Dom D 2 → E ⊗ F , extends to a selfadjoi nt r e gular op er ator with c omp act r esolvent. Mor e over, the diagr am Ψ i ( A 1 , B 1 ) × Ψ j ( A 2 , B 2 ) ✲ Ψ i + j ( A 1 ⊗ A 2 , B 1 ⊗ B 2 ) K K i ( A 1 , B 1 ) × K K j ( A 2 , B 2 ) b ❄ ⊗ ✲ K K i + j ( A 1 ⊗ A 2 , B 1 ⊗ B 2 ) b ❄ BIV ARIANT K -THEOR Y AND CORRESPONDENCES 13 c ommutes. Consequently , w e can deﬁne the externa l pro duct in this wa y , using unbounded mo dules. In [21], K ucerovsky gives suﬃcient co nditio ns for an un bo unded mo dule ( E ˜ ⊗ A F , D ) to b e the int ernal pro duct of ( E , S ) and ( F , T ). F o r each e ∈ E , we hav e an op erator T e : F → E ˜ ⊗ B F f 7→ e ⊗ f . Its adjoint is given by T ∗ e ( e ′ ⊗ f ) = h e, e ′ i f . Kucerovsky’s result now reads a s follows. Theorem 2 .2.3 ([2 1 ]) . L et ( E ˜ ⊗ B F , D ) ∈ Ψ 0 ( A, C ) . Su ppp ose that ( E , S ) ∈ Ψ 0 ( A, B ) and ( F , T ) ∈ Ψ 0 ( B , C ) ar e such that • F or e in some dense subset of A E , the op er ator  D 0 0 T  ,  0 T e T ∗ e 0  , is deﬁne d on Dom ( D ⊕ T ) and extends to an op er ator in End ∗ C ( E ˜ ⊗ B F ⊕ F ); • Dom D ⊂ Dom S ˜ ⊗ 1 ; • F or some κ ∈ R , h S x, D x i + h D x, S x i ≥ κ h x, x i for al l x in the do main. Then ( E ˜ ⊗ B F , D ) ∈ Ψ 0 ( A, C ) r epr esents the internal Kasp ar ov pr o duct of ( E , S ) ∈ Ψ 0 ( A, B ) and ( F , T ) ∈ Ψ 0 ( B , C ) . This theorem only gives suﬃcient conditions, and gives an indication abo ut the actual fo rm of the pro duct of tw o given cycles. By eq uipping un b o unded bimo dules with s ome extra diﬀer ent ial structure, we will obtain an algebraic description of the pr o duct cycle. T o this end, we need to extend o ur scop e fro m C ∗ -mo dules to a class of similar ob jects, deﬁned ov er a la rger class of top ological a lgebras . 3. Opera tor modules When dealing with unbounded o p e rators , it b ecomes necessary to dea l with dense subalgebra s of C ∗ -algebra s a nd mo dules ov er these. The theor y of C ∗ -mo dules, which is the basis of Kasparov’s approach to biv ariant K -theory for C ∗ -algebra s, needs to b e extended in an appropriate w ay . The framework of op erato r spa ces a nd the Haagerup tensor pro duct provides with a cla ss o f mo dules and algebras which is suﬃcien tly rich to accomo date for the phenomena o c curring in the Baa j-Julg picture of K K -theor y . 3.1. O p erator spaces. W e will frequently dea l with algebras a nd mo dules that are not C ∗ . In this s ection we discuss the basic notions of the theory o f op erator spa c es, in which all o f our examples will ﬁt. There is an intrinsic approach prese nt ed in [17]. The link b etw een the theory w e descr ib e here and the afo r ementioned intrinsic approach can be f ound in [27]. Deﬁnition 3.1.1. An op er ator sp ac e X is a closed linea r subspace of some C ∗ - algebra. As such there a re cano nical norms on the ma trix spaces M n ( X ) and the space K ⊗ X . A linear map φ : X → Y betw een op er ator spa ces is called c ompletely b ounde d , resp. c ompletely c ontr active , resp. c ompletely isometric if the induced map 1 ⊗ φ : K ⊗ X → K ⊗ Y , 14 BRAM M ESLAND is bounded, resp. c ontractiv e, r esp isometric for the minimal tensorpr o duct no rm. The nor m of 1 ⊗ φ is denoted k φ k cb and equals sup n k 1 n ⊗ φ k , where 1 n ⊗ φ : M n ( C ) ⊗ X → M n ( C ) ⊗ Y . An y C ∗ -mo dule E over a C ∗ -algebra B is an o p erator space, as it is is o metric to K ( B , E ), which is a close d subspace of K ( B ⊕ E ), the linking algebr a of E . Let E b e an ( A, B ) bimo dule and D an o dd regular o p e rator in E . Deﬁne A 1 := { a ∈ A : [ D, a ] ∈ End ∗ B ( E ) . } Let δ : A 1 → End ∗ B ( E ) the closed deriv a tion a 7→ [ D , a ]. Then A 1 can b e made int o an op erator space via π 1 : A 1 → M 2 (End ∗ B ( E )) (3.6) a 7→  a 0 δ ( a ) γ aγ  . (3.7) Here γ is the gr ading on E this construction in particular applies to K K - cycles for ( A, B ) ( E , D ), in which case A 1 is dense in A .Equipp ed with this op er a tor space structure, A admits a completely cont ractive alg ebra homomorphism A → End ∗ B ( G ( D )) . 3.2. The H aagerup tens o r pro duct. F or op erato r spaces X and Y , o ne can de- ﬁne their spatial t ensor pro duct X ⊗ Y as the nor m c losure of the algebraic tensor pro duct in the spatial tensor product of some containing C ∗ -algebra s. This gives rise to an e x terior tenso r pro duct of op erator mo dules. The internal tensor pro duct of C ∗ -mo dules is an e x ample of the Ha agerup ten- sor pro duct for o p erator spaces. This tensor pro duct will be extremely imp o rtant in what follows. Deﬁnition 3. 2.1. Let X , Y b e op erator space s. The Haagerup norm on K ⊗ X ⊗ Y is deﬁned by k u k h := inf { n X i =0 k x i kk y i k : u = m ( X x i ⊗ y i ) , x i ∈ K ⊗ X , y i ∈ K ⊗ Y } . Here m : K ⊗ X ⊗ K ⊗ Y → K ⊗ X ⊗ Y is the lineariz a tion of the map ( a ⊗ x, b ⊗ y ) 7→ ( ab ⊗ x ⊗ y ). Theorem 3.2.2. If X ⊂ B ( H ) and Y ⊂ B ( K ) , the norm on X ⊗ Y induc e d by the Haagerup norm is given by k X j x j ⊗ y j k h = inf {k X v i v ∗ i k 1 2 k X w ∗ i w i k 1 2 : X i v i ⊗ w i = X j x j ⊗ y j } . and the c ompletion of X ⊗ Y in this norm is a n op er ator sp ac e denote d X ˜ ⊗ Y . The completion X ˜ ⊗ Y and is called the Haagerup tensor pr o duct o f X a nd Y . F r om this theorem we deduce the following useful prop er ty . Whenev er x i ∈ X , y i ∈ Y ar e sequence s such tha t k X y ∗ i y i k , k X x i x ∗ i k ≤ 1 , then P x i ⊗ y i is conv erg ent for the Haage r up nor m and deﬁnes an ele ment w ∈ X ˜ ⊗ Y , with k w k ≤ 1. Another consequence o f this is the following. BIV ARIANT K -THEOR Y AND CORRESPONDENCES 15 Prop ositi o n 3.2 .3. F or a close d sub algebr a A ⊂ B ( H ) , op er ator multiplic ation induc es a c ompletely c ontr active ma p m : A ˜ ⊗A → A . Pr o of. First deﬁne m o n the alg ebraic tensor pro duct via a ⊗ b 7→ a b . Then estimate k m ( n X i =1 a i ⊗ b i ) k = k n X i =1 a i b i k = kh ( a ∗ i ) n i =1 , ( b i ) n i =1 ik ≤ k ( a ∗ i ) kk ( b i ) k = k X a i a ∗ i k 1 2 k X b ∗ i b i k 1 2 , where we viewed the expression P n i =1 a i b i as a n inner pro duct of t wo vectors in the C ∗ -mo dule L n i =1 B ( H ). Since this inequality holds f or any represen tative of P n i =1 a i b i , it follows from theorem 3 .2.2 that k m ( P n i =1 a i ⊗ b i ) k ≤ k P n i =1 a i ⊗ b i k h , so m is contin uous.  Deﬁnition 3. 2.4. An op er ator algebr a is an oper ator space A which is an a lge- bra, suc h that the m ultiplication induces a c o mpletely bounded map A ˜ ⊗A → A . A (righ t) op er ator mo dule is an op era tor space M whic h is a right mo dule ov er an opera tor alge br a A , such that the module m ultiplicatio n induces a completely bo unded ma p M ˜ ⊗A → M . Note that the multiplication in the ab ove deﬁnition is only abstractly deﬁned and need not coincide with o p erator m ultiplication. How ever, in [4] it is proved that such op erator alg ebras are c o mpletely b oundedly isomorphic to a subalgebra of B ( H ) for some H . The mo dule G ( D ) ⊂ E ⊕ E from e x ample 3.6 is a (left)-oper ator mo dule over the op erator algebra A . The natural c hoice of morphisms b etw een o pe r ator modules are the co mpletely bounded mo dule maps. If E and F are op erato r mo dules ov e r an op er ator alg ebra A , we deno te the set o f these maps by Hom c A ( E , F ) . A c ountable appr oximate un it for an op erato r algebr a A is a sequence { u n } ⊂ A such that sup n k u n k cb < ∞ and lim n →∞ k au n − a k = lim n →∞ k u n a − a k = 0 , for a ll a ∈ A . W e use the completely b ounded version of op era tor alg ebras a nd mo dules as the co mpletely con tractive picture is too restrictiv e for our purpo s es. Surprisingly , the cb-theor y is more complicated than the contractive theory , in so me asp ects, esp ecially when dealing with nonunital alg ebras. An excellent refer ence for op erator alg ebra and mo dule theory is [8]. Deﬁnition 3.2.5. Let A , B b e an op era tor alg ebras. A c ompletely b ounde d ant i- isomorphi sm is an antilinear bijection φ : A → B such that φ ( ab ) = φ ( b ) φ ( a ), for which the no rms of the matrix extensio ns φ ( a ij ) := ( φ ( a j i )) a re uniformly b ounded. An involutive o p er ator algebr a is an op era tor algebra which carries an in v olution a 7→ a ∗ , which is a co mpletely b ounded anti-isomorphism. If M is a rig ht- and N a left op erator mo dule over an inv o lutive op er a tor alg ebra A , a c ompletely b ounde d anti-isomorphism is an antilinear bijection φ : M → N suc h that φ ( ma ) = a ∗ φ ( m ). 16 BRAM M ESLAND Of course, C ∗ -algebra s and -mo dules are examples that ﬁt t his deﬁnition. The algebra A 1 from exa mple 3 .6 is an inv o lutive op er a tor a lgebra since π 1 ( a ∗ ) = v π 1 ( a ) ∗ v ∗ , a nd hence k a k = k a ∗ k . Now supp os e M is a righ t op erator A -mo dule, and N a le ft opera tor A -module. Denote by I A ⊂ M ˜ ⊗ N the closur e of the linear span of the expr essions ( ma ⊗ n − m ⊗ an ) . The mo dule Haag erup tensor pr o duct of M and N ov er A ([9]) is M ˜ ⊗ A N := M ˜ ⊗ N/ I A , equipp e d with the quotient norm, in whic h it is ob viously complete. Moreov er , if M also ca r ries a left B o pe rator mo dule structure, and N a right C oper ator mo dule structure, then M ˜ ⊗ A N is an operator B , C -bimo dule. Graded oper ator algebras and -mo dules can b e deﬁned by the same conv ent ions as in deﬁnition 1.1.2 and the discussion preceeding it. If the modules and ope rator algebras a r e g raded, so are the Haagerup tensor products, again in the same wa y as in the C ∗ -case, a s in the discussion around equation 1.1. The following theorem resolves the ambiguit y in the notation for the in terior tenso r pro duct of C ∗ -mo dules a nd the Haager up tensor pro duct of op era tor spaces . Theorem 3.2. 6 ([6]) . L et E , F b e C ∗ -mo dules over the C ∗ -algebr as B and C r esp e ctively, and π : B → End ∗ C ( F ) a nonde genr ate *-homo morphism. Then the interior tensor pr o duct and t he Haagerup tensor pr o duct of E and F ar e c ompletely isometric al ly isomorph ic. This result pr ovides us with a c onv enient description of algebra s of compact op erator s on C ∗ -mo dules. The dual mo dule o f a C ∗ -mo dule E is anti-isomorphic to E as a linear space, and we equip it with a left C ∗ - B -mo dule structure using the involution: be := eb ∗ , ( e 1 , e 2 ) 7→ h e 1 , e 2 i ∗ . Theorem 3.2.7 ([6 ]) . Ther e is a c ompletely isometric i somorphism K C ( E ˜ ⊗ F ) ∼ − → E ˜ ⊗ B K C ( F ) ˜ ⊗ B E ∗ . In p articular K B ( E ) ∼ = E ˜ ⊗ B E ∗ . 3.3. Stably rig ged m o dules . The work of Blecher [6] provides a metric desc r ip- tion of C ∗ -mo dules which is useful in extending the theory to non C ∗ -algebra s. The algebra K B ( E ) asso ciated to a Z / 2- g raded co un tably generated C ∗ - B -mo dule E , admits an approximate unit { u n } n ∈ N consisting of element s in Fin B ( E ). Replacing u n by u ∗ n u n if necessary , we may assume (3.8) u n = X 1 ≤| i |≤ n x i ⊗ x i , by inv oking Kasparov’s sta bilization theorem. F or each n w e get op era tors φ n ∈ K B ( E , B 2 n ), deﬁned by (3.9) φ n : e 7→ ( h x α i , e i ) 1 ≤| i |≤ n . W e hav e (3.10) φ ∗ n : ( b i ) n i = − n 7→ X 1 ≤| i |≤ n x i b i , BIV ARIANT K -THEOR Y AND CORRESPONDENCES 17 and hence φ ∗ n ◦ φ n → id E po int wise. This structure deter mines the E co mpletely as a C ∗ -mo dule. Theorem 3.3. 1 ([6]) . L et B b e a gr ade d sep ar able C ∗ -algebr a and E b e an op er ator sp ac e which is also a gr ade d right op er ator mo dule over B . Then E is c ompletely isometric al ly isomorphic to a c ountably gener ate d C ∗ -mo dule if and only if t her e exist c ompletely c ontr active mo dule maps φ n : E → B 2 n , ψ n : B 2 n → E , of de gr e e 0 , such t hat ψ n ◦ φ n c onver ges p ointwise to the identity on E . In this c ase the inner pr o duct on E is given by h e, f i = lim n →∞ h φ n ( e ) , φ n ( f ) i . F o r this reason we can think of C ∗ -mo dules a s appr oximately ﬁnitely generated pro jective mo dules. Also note that the maps φ n , ψ n are by no means unique, and that diﬀeren t maps can th us give ris e to the s ame inner pro duct on E . The description of C ∗ -mo dules in theorem 3.3.1 is metric, a nd he nc e generalize s to no n- selfadjoint op er ator alg ebras with contractive approximate unit. Deﬁnition 3.3 .2 (cf. [5 ]) . Let B be an op erato r algebra with completely con- tractive approximate identit y , and E a right B - op erator module. E is a countably generated B - rigge d mo dule if there exist co mpletely contractive B -mo dule maps φ n : E → B 2 n , ψ n : B 2 n → E , such that ψ n ◦ φ n → id E strongly o n E . Subsequently deﬁne the dual mo dule of E by E ∗ := { e ∗ ∈ Hom c B ( E , B ) : e ∗ ◦ ψ n ◦ φ n → e ∗ } , and the algebra of B - c omp act op er ators as K B ( E ) := E ˜ ⊗ B E ∗ . Remark 3 .3.3. In [5], three more conditions app ear in the deﬁnition of rigged mo dule. The ﬁrst one is tha t the mo dule E be essential , i.e. E B is dense in E . Moreov er it w as required that φ n ψ k φ k → φ n and ψ n u i → ψ n in norm. Here u i is a bounded appro ximate iden tity for B . All of these conditions w ere shown to be sup e rﬂuous in [7]. Remark 3.3.4. It is immediate from this deﬁnition that E ∗ = K B ( E , B ). This mo dule satis ﬁe s the transp osed version o f 3.3.2, i.e. it is a left rig g ed B -mo dule [5]. The mo dule structure comes fro m the left mo dule structur e on B itself, ( be ∗ )( e ) = be ∗ ( e ). F or the rigge d structure on a C ∗ -mo dule, co ming from the approximate unit (3.8), the structural maps ψ ∗ n : E ∗ → ( B 2 n ) t and φ ∗ n : ( B 2 n ) t → E ∗ are given by ψ ∗ n ( e ∗ ) := ( e ∗ ( x i )) t 1 ≤| i |≤ n , φ ∗ n ( b i ) t 1 ≤| i |≤ n := X 1 ≤| i |≤ n b i x i . There is a n a nalogue of adjointable op er ators on rigg ed mo dules. Their deﬁnition is stra ightforw a rd. Deﬁnition 3.3.5 ([5]) . A completely b ounded op era tor T : E → F b etw een rigg ed mo dules is called adjoi ntable if there exists an op era tor T ∗ : F ∗ → E ∗ such that ∀ e ∈ E , f ∗ ∈ F ∗ : h f ∗ , T e i = h T ∗ f ∗ , e i . Here we use d th e suggestive notation h f ∗ , T e i for f ( T e ). The space of adjointable op erator s fro m E to F is denoted End ∗ B ( E , F ). 18 BRAM M ESLAND When B has a contractiv e appro ximate unit it is a rigged module over itself, and K B ( B ) ∼ = B completely iso metrically . The c o mpact and adjoin table operator s satisfy the usual relation End ∗ B ( E ) = M ( K B ( E )), where M denotes the multiplier algebra. W e take this as the deﬁnition o f M ( B ). Given an op er ator algebra and a completely con tr a ctive algebra homomorphism A → End ∗ B ( E ), E is an ( A , B ) rigged bimo dule. As can b e e xp e cted from theorem 3 .2.6, the Haag erup tensor pro duct of rigg ed mo dules behaves like the in terior tensor pro duct of C ∗ -mo dules. Theorem 3.3.6 ([5]) . Le t E b e a right B - rigge d mo dule and F an ( B , C ) ri gge d bimo dule. Then E ˜ ⊗ B F is a C - rigge d mo dule and K C ( E ˜ ⊗ B F ) ∼ = E ˜ ⊗ B K C ( F ) ˜ ⊗ B E ∗ c ompletely isometric al ly. F o r our pur p o ses, we ar e only considered with countably generated C ∗ -mo dules. The par ticular form o f the approximate un it (3 .8) implies the maps φ n , ψ n from (3.9) ca n b e assembled int o tw o maps φ : E → H B , ψ : H B → E , given by φ ( e ) = ( h x i , e i ) i ∈ Z and ψ ( b i ) i ∈ Z = P i ∈ Z x i b i . Then we have ψ φ = id and φψ is a pro jection. In [5], these mo dules are called CCGP (countably column generated pro jective) mo dules. As noted by Blecher in [5], rigged mo dules seem to o r estrictive for K -theoretic consider ations, as it is unlikely that every ﬁnite pro- jective mo dule o ver a n operato r a lgebra may be rigged. How ever, if we allow the maps φ n , ψ n from deﬁnition 3 .3.2 to b e completely bo unded, w e obtain a theory that is ﬂexible enough. Let H := ℓ 2 ( Z \ { 0 } ) ∼ = ℓ 2 ( N ) ⊕ ℓ 2 ( N ) be an inﬁnite dimensio na l separable graded Hilber t column space and B a graded op erato r algebra. Then the H B := H ˜ ⊗B is the st andar d rigge d mo dule over B . Deﬁnition 3.3. 7 . A right B op erator mo dule E is stably rigge d if there are com- pletely b ounded maps φ : E → H B and ψ : H B → E suc h that ψ φ = id . A stably rigged module need not be rigg ed itself. This will be the cas e if φ, ψ can b e chosen co mpletely c o ntractiv e . F or this reaso n, we will always consider stably rigged mo dules up to cb-isomorphism . In general a stably rig ged mo dule is a completely bounded dir ect summand in H B , which is an actual rigged module. The maps φ n , ψ n deﬁned b y comp os ing φ and ψ with the pr o jections H B → B n and inclusions B n → H B will b e uniformly completely b ounded as opp os ed to co mpletely contractiv e. They ca n be used to deﬁne the algebr a s K B ( E ) a nd End ∗ B ( E ) a s above. In the prese nc e of a countable approximate unit { u n } ⊂ B , B is sta bly rigged ov er itself a nd K B ( B ) ∼ = B co mpletely boundedly . Deﬁnition 3.3.8. The multiplier algebr a of an op erator algebra B with a countable approximate unit is M ( B ) := E nd ∗ B ( B ). Note that this deﬁnes M ( B ) up to cb-isomor phism, which suﬃces for our pur- po ses. Theorem 3.3 .9. L et E b e a stably rigge d B -mo dule and F a st ably rigge d C mo dule. Given a c ompletely b ounde d algebr a homomorphism π : B → End ∗ C ( F ) , the H aagerup tensor pr o duct E ˜ ⊗ B F is a stably rigge d mo dule a nd K C ( E ˜ ⊗ B F ) ∼ = E ˜ ⊗ B K C ( F ) ˜ ⊗ B E ∗ c ompletely b ounde d ly. Mor e over, if C = C is a C ∗ -algebr a, then b oth F and E ˜ ⊗ B F ar e c ompletely iso morphic to C ∗ -mo dules. BIV ARIANT K -THEOR Y AND CORRESPONDENCES 19 Pr o of. W e o nly prove the la st statement. First note that the mo dule F is completely isomorphic to p H C , with p = φψ an idemp otent in End ∗ C ( H C ), which is a C ∗ - mo dule. Secondly , denote by ˜ B the algebr a B with the completely iso morphic op erator space structure given by the representation id ⊕ π : B → B ⊕ E nd ∗ C ( F ) . Then π : ˜ B → E nd ∗ C ( F ) is completely con tractive, and H ˜ B remains rig ged for this op erator space structure, and is completely isomorphic to H B . Thu s, w e see that E ˜ ⊗ B F is completely isomorphic to a C ∗ -mo dule, by theor em 3.3 .6.  The nex t theor em shows that the Haa gerup tensor pro duct of stably r ig ged mo dules b ehaves well with resp ect to adjointable op era tors. Theorem 3.3. 10 (cf. [5]) . L et E , E ′ b e stably rigge d B -mo dules, F , F ′ stably rigge d ( B , C ) - bimo dules. If S ∈ End ∗ B ( E , E ′ ) , T ∈ End ∗ C ( F, F ′ ) , and T is also a left B - mo dule map, then S ⊗ T ∈ End ∗ C ( E ˜ ⊗ B F, E ′ ˜ ⊗ B F ′ ) . Mor e over the map S 7→ S ⊗ 1 is a c ompletely b ounde d alge br a h omomorphism. The direct sum of a family { E α } of r igged mo dules is cano nically deﬁned in [5]. This is do ne by em b edding the alg ebra B iso metrically in a C ∗ -algebra B . The mo dules E α are completely isometrically isomorphic to clo sed s ubmo dules of the C ∗ -mo dules E α := E α ˜ ⊗ B B , a nd L E α is constructed as the natural c lo sed submo dule of the C ∗ -direct sum L α E α . F or stably rigged m o dules the situation is slig htly more complicated. Deﬁnition 3.3.11. L e t { E α } α ∈ N be a co untable family of stably rigged mo dules, with s tr uctural maps φ α : E α → H B and ψ α : H B → E α . Suppo se sup α {k ψ α k cb , k φ α k cb } < ∞ . The direct sum E := L α E α is deﬁned up to cb-ismor phism by identifying it with the s ubmo dule L α φ α ( E α ) ⊂ L α H B . The maps φ := M φ α : M α E α → H B , ψ := M ψ α : H B → M α E α , give a completely b ounded factoriza tion o f the iden tity , making E into a s tably rigged mo dule. Note that we used the isomorphis m L α ∈ N H B ∼ = H B in the deﬁnition of the maps φ, ψ . The c hoice of op erator spa ce structure on the direct sum is na tur al, but that it is more natural to think ab out the direct sum as b eing deﬁned only up to complete iso morphism. F o r o ur purp o s es this suﬃces. 4. Smoothness W e adopt the philosophy tha t sp ectral triples should b e a source o f smo oth structures C ∗ -algebra s. The mo s t imp o r tant feature of a smo oth subalge br a is stability under holomorphic fun ctional calculus, implying K -equiv alence. W e will show our smo oth algebr as satisfy this prop erty . Moreover, w e sho w that regular sp ectral triples [11] ar e smoo th in our sense , providing us with nu merous examples. Subsequently , we turn to the no tion of a smo oth C ∗ -mo dule over a C ∗ -algebra 20 BRAM M ESLAND equipp e d with a smo oth structure. All oper ator alg ebras are assumed to have a completely b ounded countable approximate unit. 4.1. So b olev algebras. W e construct now a nested sequence o f algebr as · · · ⊂ A i +1 ⊂ A i ⊂ A i − 1 ⊂ · · · ⊂ A 1 ⊂ A, for any gra ded ( A, B )-bimo dule E equipp ed with an o dd selfadjoint reg ular op erator D . Each A i will admit a co mpletely co ntractive repr esentation on the i -th Sob olev mo dule of D . The represent ation π 1 : A 1 → M 2 (End ∗ B ( E )) (equation 3.6), asso cia ted to a n ( A, B )-bimo dule E equipped with an o dd regular op erator D , induces a represen- tation A 1 → End ∗ B ( G ( D )) a 7→ pπ ( a ) p, with p = p D the W oronowicz pr o jection. This is an alg e bra homomorphism due to the identit y pπ 1 ( a ) p = π 1 ( a ) p . F r om this it follows that A 1 → End ∗ B ( v G ( D )) a 7→ p ⊥ π ( a ) p ⊥ , where p ⊥ := 1 − p , is a ho momorphism as well. Th us we can deﬁne a map θ 1 : A 1 → M 2 (End ∗ B ( E )) a 7→ pπ 1 ( a ) p + p ⊥ π 1 ( a ) p ⊥ . Recall from the discussion pr e ceding prop ositio n 1.3 .4, that the natural gra ding to consider on L 2 i +1 j =1 E is deﬁned inductively by γ i +1 :=  γ i 0 0 − γ i  . Deﬁnition 4.1.1. Let A 1 , π 1 and θ 1 be as ab ove. F or i > 0 , abusively deno te by D the o dd selfadjoint regular op erator on L 2 i j =1 E given by the diagona l action of D , and b y p i its W oro nowicz pro jection. F or i < k , p i will denote the corr esp onding diagonal ma tr ix in L 2 k j =1 E . Inductively deﬁne (4.11) A i +1 := { a ∈ A i : [ D, θ i ( a )] ∈ E nd ∗ B ( 2 i M j =1 E ) } , π i +1 : A i +1 → M 2 i +1 (End ∗ B ( E )) (4.12) a 7→  θ i ( a ) 0 [ D , θ i ( a )] γ i θ i ( a ) γ i  , θ i +1 : A i +1 → M 2 i +1 (End ∗ B ( E )) (4.13) a 7→ p i +1 p i π i +1 ( a ) p i p i +1 + p ⊥ i +1 p ⊥ i π i +1 ( a ) p ⊥ i p ⊥ i +1 The notion of smoo thness in tro duced in the next section will entail that the A i ’s are dense in A . In the curre nt section, no such assumption is pres ent . W e will refer to A i as the i -th Sob olev sub algebr a of A . In case A = End ∗ B ( E ), w e denote the i -th ful l Sob olev algebr a o f D b y Sob i ( D ). Clear ly , A i = A ∩ Sob i ( D ). BIV ARIANT K -THEOR Y AND CORRESPONDENCES 21 Remark 4.1.2. Note that we ha ve deﬁned π i and θ i on the sa me domain A i . How ever, a pr io ri, we hav e Dom π i , Dom π i +1 ⊂ Dom θ i ⊂ E nd ∗ B ( E ) . It is imp o rtant to think of these representations in this wa y when one consider s density of the domains. T a king π 0 to be the original r epresentation o f A on E , the direct s ums (4.14) π [ i ] = i M j =0 π j : A i → i M j =0 End ∗ B ( 2 i M k =1 E ) give each A i the structure of an operator spa ce, and this yields an in verse sys tem of op era tor alg e bras · · · → A i +1 → A i → A i − 1 → · · · → A 1 → A, in which all maps are completely contractive. Consider the unitar ie s v n +1 :=  0 − I 2 n I 2 n 0  ∈ E nd ∗ B ( 2 n +1 M j =1 E ) , where I 2 n is the 2 n × 2 n -identit y ma trix. F o r j < i we identify v i with v i I 2 j and as such consider it as an element of End ∗ B ( L 2 j i =1 ). As such, v i and v k commute for all k , i ≤ j . F or i ∈ N , denote by [ i ] the set { 1 , · · · , i } and b y P ([ i ]) the pow er set of [ i ]. Deﬁne v F := Y j ∈ F v j ∈ M 2 i (End ∗ B ( E )) , which is well deﬁned s ince the v j commute. Note that v [0] = v ∅ = 1 . Prop ositi o n 4. 1 .3. The A i ar e invo lutive op er ator algebr as. Pr o of. T o prove that the in volution a 7→ a ∗ is a complete an ti isometry for the norm k · k i (cf. deﬁnition 3 .2.4) we show that (4.15) π i ( a ∗ ) = v [ i ] π i ( a ) ∗ v ∗ [ i ] , θ i ( a ∗ ) = v [ i ] θ i ( a ) ∗ v ∗ [ i ] , i even; (4.16) π i ( a ∗ ) = v [ i ] γ i π i ( a ) ∗ γ i v ∗ [ i ] , θ i ( a ∗ ) = v [ i ] γ i θ i ( a ) ∗ γ i v ∗ [ i ] , i o dd . In order to a chiev e this, reca ll that the g rading on E nd ∗ B ( L 2 i j =0 E ) is given by T 7→ γ i T γ i , and hence that [ D , T ] = D T − γ i T γ i D . F ro m this, it is immediate that ( γ i T γ i ) ∗ = γ i T ∗ γ i , [ D, T ] ∗ = γ i [ D , T ∗ ] γ i = − [ D , γ i T ∗ γ ] , which will b e used in the c o mputation below. W e have v [0] = 1 and the v i commute with D . F or π 0 = θ 0 , 4.15 is tr ivial. Suppose 22 BRAM M ESLAND 4.15 holds for some even num b er i . Then, π i +1 ( a ∗ ) = v [ i ] θ i ( a ) ∗ v ∗ [ i ] 0 [ D , v [ i ] θ i ( a ) ∗ v ∗ [ i ] ] v [ i ] γ i θ i ( a ) ∗ γ i v ∗ [ i ] ! =  0 − v [ i ] v [ i ] 0   γ i θ i ( a ) ∗ γ i − [ D , θ i ( a ) ∗ ] 0 θ i ( a ) ∗  0 v ∗ [ i ] − v ∗ [ i ] 0 ! =  0 − v [ i ] v [ i ] 0   γ i θ i ( a ) γ i 0 − γ i [ D , θ i ( a )] γ i θ i ( a )  ∗ 0 v ∗ [ i ] − v ∗ [ i ] 0 ! = v [ i +1] γ i +1 π i +1 ( a ) ∗ γ i +1 v ∗ [ i +1] . Since v [ i ] commutes with D , w e ha ve v [ i +1] p i +1 p i v ∗ [ i +1] = p ⊥ i +1 p ⊥ i , and the pr o jec- tions p i , p i +1 are ev en. Th us , 4 .16 holds for i + 1. Now supp ose 4.1 6 holds for some o dd i . Note that for all i , γ i v [ i ] = ( − 1 ) i γ i v [ i ] , i.e. v [ i ] is homog eneous of degree i mo d 2. Then, π i +1 ( a ∗ ) = v [ i ] γ i θ i ( a ) ∗ γ i v ∗ [ i ] 0 [ D , v [ i ] γ i θ i ( a ) ∗ γ i v ∗ [ i ] ] v [ i ] θ i ( a ) ∗ v ∗ [ i ] ! =  0 − v [ i ] v [ i ] 0   θ i ( a ) ∗ γ i [ D , θ i ( a ) ∗ ] γ i 0 γ i θ i ( a ) ∗ γ i  0 v ∗ [ i ] − v ∗ [ i ] 0 ! =  0 − v [ i ] v [ i ] 0   θ i ( a ) 0 [ D , θ i ( a )] γ i θ i ( a ) γ i  ∗ 0 v ∗ [ i ] − v ∗ [ i ] 0 ! = v [ i +1] π i +1 ( a ) ∗ v ∗ [ i +1] . Since v [ i ] commutes with D , w e have v [ i +1] p i +1 p i v ∗ [ i +1] = p ⊥ i +1 p ⊥ i , a nd he nce it follows that 4.15 ho lds for i + 1 .  Prop ositi o n 4. 1 .4. F or e ach n ∈ N , ther e is a de c omp osition (4.17) 2 n M i =1 E ∼ = M F ∈ P ([ n ]) v F G ( D n ) , and fo r a ∈ A n , θ n ( a ) r esp e cts this de c omp osition. In fact it is nonzer o only on G ( D n ) and v [ n ] G ( D n ) . Pr o of. The decomp osition is proved b y induction. Clearly it holds for n = 1 (this is W o ronowicz’s theor e m 1.5). Supp ose we have the decomp ositio n for n = k . Then 2 k +1 M i =1 E ∼ = M F ∈ P ([ k ]) v F G ( D k ) ⊕ M F ∈ P ([ k ]) v F G ( D k ) , and since v F ( G ( D k ) ⊕ G ( D k )) ∼ = v F ( G ( D k +1 ) ⊕ v k +1 G ( D k +1 )) , we get the desir ed dec omp osition for n = k + 1. T o prove the A n -inv ariance, observe that for n = 1, this holds by cons tr uction. Suppos e the statement has b een prov en for n = i . The graph of D as a diago nal op erator in L 2 i i =1 E is a submo dule of L 2 i +1 i =1 E and under the iso morphism 4.17 it ge ts mapp ed to L F ∈ P ([ i +1]) v F G ( D i +1 ). BIV ARIANT K -THEOR Y AND CORRESPONDENCES 23 Thu s, preserv ation of the decomp o s ition 4.17 is equiv alent to preser v ation of the graph of D and its co mplemen t. This is immediate from the deﬁnition of θ i +1 .  Corollary 4.1.5. Each A n admits a c ompletely c ontr active r epr esentation χ n : A n → End ∗ B ( G ( D n )) . Pr o of. Denote by p [ n ] = Q n i =1 p i ∈ End ∗ B ( L 2 n i =1 E ) the pro jection onto G ( D n ). F r om the previous prop osition it follows that χ n ( a ) := p [ n ] θ n ( a ) p [ n ] = θ n ( a ) p [ n ] , and hence is a completely contractive algebr a ho momorphism.  Note that in fact w e hav e θ n ( a ) = p [ n ] θ n ( a ) p [ n ] + v [ n ] p [ n ] v ∗ [ n ] θ n ( a ) v [ n ] p [ n ] v ∗ [ n ] for even n , and θ n ( a ) = p [ n ] θ n ( a ) p [ n ] + v [ n ] p [ n ] v ∗ [ n ] γ n θ n ( a ) γ n v [ n ] p [ n ] v ∗ [ n ] for o dd n . Corollary 4.1 .6. a ∈ A n +1 if and only if a ∈ A n and [ D, χ n ( a )] , [ D , χ n ( a ∗ )] ∈ End ∗ B ( L 2 n i =1 E ) . Pr o of. W e hav e χ n ( a ∗ ) = p [ n ] v [ n ] θ n ( a ) ∗ v ∗ [ n ] p [ n ] , for even n and χ n ( a ∗ ) = p [ n ] v [ n ] γ n θ n ( a ) ∗ γ n v ∗ [ n ] p [ n ] , for o dd n . Therefore , for o dd n k [ D , θ n ( a )] k = ma x {k [ D , χ n ( a )] k , k v [ n ] [ D , p [ n ] v ∗ [ n ] γ n θ n ( a ) γ n v [ n ] p [ n ] ] v ∗ [ n ] k} = max {k [ D , χ n ( a )] k , k [ D , χ n ( a ∗ )] k} . The same works fo r even n .  Lastly , we note that the constructions asso cia ted with Sob o lev alge br as can be done for almo st selfadjoint opera tors, using the nonse lfadjoint idempo tent s from corolla r y 1.3.8. The price for doing this is that the in volution will not b e com- pletely isometric, but still a complete anti isomor phism. This is g o o d enough for our pur po ses, and ﬁts the idea of working with nonselfadjoint algebra s and homo- morphisms. 4.2. H o lomorphic stability . Now we turn to sp ectral inv ariance of the A i . The following deﬁnition is a mo diﬁca tion of [3], deﬁnition 3 .1 1: Deﬁnition 4. 2.1. Let A be an algebr a with Banach norm k · k , and A its closure in this norm. A norm k · k α on A is said to be a nalytic with r e s pe ct to k · k if for each x ∈ A , with k x k < 1 we hav e lim sup n →∞ ln k x n k α n ≤ 0 . The reason for introducing the concept of a nalyticity is tha t analytic inclusions are sp ectra l inv ar ia nt. Prop ositi o n 4.2.2 ([3]) . L et A β → A α b e a c ontinuous dense inclusion of un ital Banach algebr as. If k · k β is analytic with r esp e ct to k · k α , t hen for al l a ∈ A β we have Sp β ( a ) = Sp α ( a ) . 24 BRAM M ESLAND Pr o of. It suﬃces to show tha t if x ∈ A β is in v ertible in A α , then x − 1 ∈ A β . T o this end choos e y ∈ A β with k x − 1 − y k < 1 2 k x k α . Then k 2 − 2 xy k α < 1. By analyticity , there exists n such that k (2 − 2 xy ) n k β < 1 , and hence 2 / ∈ Sp β (2 − 2 xy ). But then 0 / ∈ Sp β (2 xy ), hence 2 xy has a n inv ers e u ∈ A β . Therefore x − 1 = 2 y u .  In order to prove sp ectral in v a riance of t he inclusions A i +1 → A i we need the following straightforw ard result, whose pro o f w e include for the sa ke of complete- ness. Lemma 4.2 .3. L et A b e a gr ade d Banach algebr a and δ : A α → M a densely deﬁne d close d gr ade d derivation into a Banach A -bimo dule M . Then k a k α := k a k + k δ ( a ) k is anal ytic with r esp e ct to k · k . Pr o of. Let k x k < 1 . W e hav e k δ ( x n ) k ≤ n k δ ( x ) k , by an obvious induction. Then lim sup n →∞ ln k x n k α n = lim sup n →∞ ln( k x n k + k δ ( x n ) k ) n ≤ lim sup n →∞ ln(1 + n k δ ( x ) k ) n ≤ lim sup n →∞ ln n n + ln(1 + k δ ( x ) k ) n = 0 .  Theorem 4.2. 4. Le t ( E , D ) b e an unb ounde d ( A, B ) bimo dule. Then al l inclusions A i +1 → A i ar e sp e ctr al invariant, and h enc e the A i ar e stable under holomorphic functional c alculus in A . Pr o of. Obse rve that k a k i +1 ≤ k a k i + k [ D , θ i ( a )] k , th us, by lemma 4.2.3 , k · k i +1 is ma jorized by a norm analytic with resp ect to k · k i , and hence is itself ana lytic with res pec t to k · k i . No w A 1 is dense in its C ∗ -closure which is a C ∗ -subalgebra of A , so A 1 is spectral in v a r iant in A . Supp ose no w A i is sp ectral inv ar iant in A . By the ab ov e arg ument , A i +1 is sp ectra l inv ariant in its i -closur e, which is sp e c tral inv a riant in A .  Corollary 4.2. 5 . L et q ∈ A k b e an idemp otent and p := q q ∗ (1 + ( q − q ∗ )( q ∗ − q )) − 1 . Then p ∈ A k and p = p 2 = p ∗ is a pr oje ction such that pq = q and q p = p . In p articular q A k = p A k . Pr o of. The element ( q − q ∗ )( q ∗ − q ) = ( q − q ∗ )( q − q ∗ ) ∗ is p ositive and hence x = 1 + ( q − q ∗ )( q ∗ − q ) is inv ertible. By theorem 4 .2.4 x − 1 ∈ A k and thus p ∈ A k . W e hav e q q ∗ x = q q ∗ (1 + ( q − q ∗ )( q ∗ − q )) = ( q q ∗ ) 2 = (1 + ( q − q ∗ )( q ∗ − q )) q q ∗ = xq q ∗ , so q q ∗ x − 1 = x − 1 q q ∗ , w hich shows that p ∗ = p and also p 2 = ( q q ∗ ) 2 x − 2 = q q ∗ xx − 2 = q q ∗ x − 1 = p. The identit y q p = p is immediate, and pq = (1+ q q ∗ + q ∗ q − q ∗ − q ) − 1 q q ∗ q = (1 − (1+ q q ∗ + q ∗ q − q ∗ − q ) − 1 )(1+ q ∗ q − q ∗ − q ) q = q . BIV ARIANT K -THEOR Y AND CORRESPONDENCES 25  In t he sequel, b y a C k -structu r e on a C ∗ -algebra A we shall mea n an in verse system o f op era tor algebras A k → A k − 1 → · · · → A where the ma ps ar e sp ectra l inv ar iant completely b ounded * - homomorphisms with dense ra nge. Deﬁnition 4. 2.6. Let A and B b e C ∗ -algebra s, E b e an ( A, B ) bimo dule, D a selfadjoint r egular opera tor in E and k > 0. The pair ( E , D ) is said to be C k if the subalge br a A k (4.11) is dense in A a nd there is a countable p ositive increa sing approximate unit u n such that sup n k u n k k < ∞ . The bimo dule ( E , D ) is smo oth if it is C k for all k . A C k -algebr a shall b e a C ∗ -algebra equipp ed with a ﬁxed C k -sp ectral triple. As such it has a natura l C k -structure. Note that if a mo dule is C k for so me k , then it is C i for a ll i ≤ k . In pa rticular K K -cycles a r e C 1 by deﬁnition. The above notion of s mo othness is weaker than the one deﬁned [11]. W e refer to the a ppe ndix for a pro of of this. In what follows (esp e cially in s ection 6 ) it is c rucial that we work rela tive to a ﬁxed sp ectral triple. Notice the par allel with the deﬁnition of a manifold a s a top olo g ical space with extra structur e deﬁned on it. 4.3. Sm o oth C ∗ -mo dul es. W e will deﬁne C k -structures on C ∗ -mo dules o v er a C k -algebra by requiring the existence of an appropriate approximate unit. W e use this to construct a chain of stably r igged submo dules E k ⊂ E k − 1 ⊂ · · · ⊂ E 1 ⊂ E , up to the smo othness deg r ee of the mo dule. Then w e show that the smo oth struc- ture is compa tible with tensor pro ducts, and we address the case of nonunital algebras . Deﬁnition 4.3.1. Let B be a smo oth C ∗ -algebra , with smo oth s tructure { B i } . A C ∗ - B -mo dule E is a C k - B -mo dule, if there is an approximate unit u n := X 1 ≤| i |≤ n x i ⊗ x i ∈ Fin B ( E ) , with x i homogeneous elemen ts s uch that the matrices ( h x i , x j i ) ∈ M n ( B k ), a nd k ( h x i , x j i ) k k ≤ C k . It is a smo oth C ∗ -mo dule if there is such an approximate unit that makes it a C k -mo dule for all k . F r om this deﬁnition, the deﬁnition o f a non unital smoo th C ∗ -algebra is forced. In order that B be smooth ov er itself, t he existence of a positive, cont ractive ap- proximate unit that restricts to a b ounded one in ea ch B k is required. This is in line with deﬁnition 4.2.6. Prop ositi o n 4.3.2 . L et B b e a C k -algebr a and E a smo oth C ∗ - B -mo dule, with c orr esp onding ap pr oximate unit u n := P 1 ≤| i |≤ n x i ⊗ x i . Then E k := { e ∈ E : h x i , e i ∈ B k , k ( h x i , e i ) i ∈ Z k k < ∞} , 26 BRAM M ESLAND is a st ably rigge d B k -mo dule. When C ≤ 1 , it is an actu al rigge d mo dule. Mor e- over, the inclusions E k +1 → E k ar e c ompletely c ontr active with dense r ange, and E k +1 ˜ ⊗ B k +1 B k ∼ = E k , c ompletely b ounde d ly. When C ≤ 1 , t his isomorphism is c ompletely isometric. Pr o of. The maps φ : E k → H B k e 7→ ( h x i , e i ) i ∈ Z \{ 0 } , and ψ : H B k → E k ( b i ) i ∈ Z 7→ X i ∈ Z \{ 0 } x i b i , will g ive the desired factor ization o f the iden tit y . These maps ar e completely bo unded o f norm ≤ C for the matr ix norms on E k given by k ( e ij ) k k := k ( φ k ( e ij )) k k , and E k is (b y deﬁnition) complete in these matrix no rms. T o c hec k that E k is a stably r igged- B k -mo dule, we hav e to s how that X n ≤| i |≤ m x i h x i , e i → 0 , in k -no rm, as n → ∞ . k X n ≤| i |≤ m x i h x i , e ik k = k ( X n ≤| i |≤ m h x j , x i ih x i , e i ) j ∈ Z k k = k ( h x j , x ℓ i ) j,ℓ ∈ Z ( h x i , e i ) n ≤| i |≤ m k k ≤ C k ( h x i , e i ) n ≤| i |≤ m k k → 0 , bec ause k ( h x i , e i ) i ∈ Z k k < ∞ . T o see that E k is dense in E , it suﬃces to show that all the x j are in E k , b e c ause they form a generating set for E . Thus we have to show that k x j k k < ∞ . T o this end we ma y ass ume that B k is unital, and we denote by { e j } j ∈ Z the s ta ndard ortho normal basis of H B k . k x j k k = k φ ( x j ) k k = k ( h x i , x j i ) i ∈ Z k k = k ( h x i , x ℓ i ) i,ℓ ∈ Z · e j k k ≤ k ( h x i , x ℓ i ) i,ℓ ∈ Z k k ≤ C . F o r the la st statement, the iso morphism will b e implemented by the mu ltiplication map m : E k +1 ˜ ⊗ B k +1 B k → E k e ⊗ b 7→ e b. BIV ARIANT K -THEOR Y AND CORRESPONDENCES 27 W r ite p k = φ k ψ k . The map m : E k +1 ˜ ⊗ B k +1 B k → E k ﬁts into a commutativ e diagram E k +1 ˜ ⊗ B k +1 B k ✲ p k +1 H B k +1 ˜ ⊗ B k +1 B k E k ❄ ✲ p k H B k , ❄ in which all other a r rows are complete isomorphisms.  Remark 4.3.3 . Ther e may very well be o ther a pproximate units sa tisfying deﬁni- tion 4.3 .1. They need not deﬁne the same C k -submo dules. Two C k -approximate units u n = P 1 ≤| i |≤ n x i ⊗ x i and v n = P 1 ≤| i |≤ n y i ⊗ y i are equiv ale nt if the matrix ( h x i , y j i ) has ﬁnite k -norm. In this case, u n and v n deﬁne the same C i -submo dules, i ≤ k , a nd the op era tor space top olo gies fr om pro p o sition 4 .3.2 a re c b-isomorphic. Therefore we think of the C k -submo dules up to cb-isomor phism. Now that we hav e co nstructed C k -submo dules a s s ta bly rigged modules, they come with ca nonical endomorphism a lgebras. This a llows for a deﬁnition of C k - bimo dule. Deﬁnition 4. 3 .4. Let A, B b e C k -algebra s, E ⇌ B a C k -mo dule and A → End ∗ B ( E ) a *-homomor phis m. E is a C k - ( A, B ) -bimo dule if the A -module struc- ture re s tricts to a co mpletely bounded homomor phism A k → End ∗ B k ( E k ). Note that a C k -bimo dule is automatica lly C i for i ≤ k . 4.4. Inner pro ducts, stabil ization and tens or pro ducts. F or a s mo oth C ∗ - algebra B with s mo oth str uctur e { B i } , any rig ht rig ged B i -mo dule has a canonically asso ciated left rigg ed B i -mo dule ˜ E . As a set, this is ˜ E := { e : e ∈ E } , equipp e d with the ca nonical conjugate linea r structure a nd the left mo dule structure a e := ea ∗ . The left-stably rigged structure comes from th e completely isometric anti iso mo rphism b etw een row- and column mo dules H B k → H t B k ( a j ) 7→ ( a ∗ j ) t , induced by the inv olution on B k . The structural ma ps ar e given by ˜ φ ( e ) := ( φ ( e ) ∗ j ) t = ( h e, x i i ) i ∈ Z , ˜ ψ (( b j ) t ) := ψ (( b ∗ j )) = X ∈ Z x i b ∗ i = X i ∈ Z b i x i , and ar e left-mo dule maps ha ving the desired prop erties. Lemma 4. 4.1. L et E b e a smo oth C ∗ -mo dule over a smo oth C ∗ -algebr a B with smo oth structure { B i } . Ther e is a cb-isomorphism of rigge d mo dules E i ∗ ∼ = ˜ E i given by r estriction of the inn er pr o duct p airing on E . Pr o of. The inner pro duct on E induces an injection ˜ E k → E k ∗ , whic h w e denote e 7→ e ∗ . F o r s uch elements we hav e (4.18) φ ∗ ( e ∗ ) = ( h e, x i ) i i ∈ Z = ˜ φ ( e ) . 28 BRAM M ESLAND This follows from the deﬁnition of ˜ φ and re ma rk 3.3.4. An element f ∈ E k ∗ is by deﬁnition a norm limit f = lim n →∞ n X i = − n f ( x i ) x ∗ i , and by (4.18) the sequence P n i = − n f ( x i ) x i is conv er gent in ˜ E k . Therefore the map e 7→ e ∗ is an isomo r phism.  As a consequence , C k -mo dules over a C k -algebra { B k } a r e pre- C ∗ -mo dules, i.e. they come with a nondegenerate B k -v alued innerpro duct pairing sa tis fying all the prop erties o f deﬁnition 1.1.1 . It should be noted that this inner product does not generate the op era tor s pace top o logy on E k . Completing bo th B k and E k yield the C ∗ -mo dule E ⇌ B . The type of self-duality expressed in lemma 4.4.1 allows us to re move the require- men t of complete bo undedness in the deﬁnition of adjointable op erato r (3.3.5). Theorem 4.4.2. L et B b e a C k -algebr a and E ⇌ B a C k -mo dule. If T , T ∗ : E k → E k ar e mappings satisfying h T e, f i = h e, T ∗ f i for al l e , f ∈ E i , then T , T ∗ ar e c ompletely b ounde d and B k -line ar, i.e. T , T ∗ ∈ End ∗ B k ( E k ) . Mor e over, the cb- norm and the op er ator norm ar e e quivalent to one another and T 7→ T ∗ is a wel l deﬁne d c omplete anti isomorph ism of End ∗ B k ( E k ) . Pr o of. W e ﬁrst prov e the statement for the case where E k is actually rigged. Uniqueness of the adjoin t and B k -linearity a re straightf orward to show. T o show T , T ∗ are bounded, ﬁrs t no te that lemma 4 .4.1 implies that K B k ( E k , B k ) is an ti isometric to E k via e 7→ e ∗ . Now let T , T ∗ be as stated in the theo rem, and take e ∈ E k with k e k k = 1. Then T e := ( T e ) ∗ ∈ K B k ( E k , B k ) a nd (4.19) k T e ( f ) k k = kh T e, f ik k = kh e, T ∗ f ik k ≤ k T ∗ f k k . F r om the Banach-Steinhaus theorem w e conclude that the set {k T e k k : k e k k = 1 } , is b ounded, whic h implies that k T k k < ∞ . By re versing T and T ∗ , w e ﬁnd k T ∗ k k < ∞ as well. Mor ever, now 4.19 implies that k T k ≤ k T ∗ k , and again, re versing g ives k T k = k T ∗ k . Complete b o undedness follows by estimating (cf. [5], theorem 3.5 ) k ( T e ij ) k k = lim n k ψ n φ n T ψ n φ n ( e ij ) k k ≤ (sup n k φ n T ψ n k cb ) sup n k ψ n ( e ij ) k k = (sup n k φ n T ψ n k ) k ( e ij ) k k ≤ k T kk ( e ij ) k k . Here we use d that φ n T ψ n : B 2 n k → B 2 n k is completely bounded, which fo llows fr om the fact that it comes fr om left multiplication by a matrix, since B k has a b ounded approximate unit. Note that this estimate also shows k T k cb = k T k . F o r the gener a l stably rigged case, embed E k in the rig g ed module H B k , i.e. choose an isomor phism E k ∼ = p k H B k , with p k ∈ E nd ∗ B k ( H B k ) a pr o jection. The equalities k T k = k T k cb = k T ∗ k , v alid for T ∈ End ∗ B k ( H B k ) then yield equiv alences of these three nor ms for End ∗ B k ( E k ).  BIV ARIANT K -THEOR Y AND CORRESPONDENCES 29 Note that this r esult implies that unitar y op erators (in the usual inner pr o duct sense), need no t be iso metric, but they will b e cb-iso mo rphisms. P a ssing to an equiv alent approximate unit (cf. rema rk 4.3.3) yields a unitar y isomor phis m of C k -submo dules. Theorem 4.4.3. L et B b e a smo oth gr ade d C ∗ -algebr a, and E a c ountably gener- ate d smo oth gr ade d C ∗ -mo dule. Then E ⊕ H B is C k unitarily isomorphi c to H B . That is, t her e is a u nitary isomorphism of gr ade d inverse systems · · · ✲ E i +1 ⊕ H B i +1 ✲ E i ⊕ H B i ✲ · · · ✲ E ⊕ H B · · · ✲ H B i +1 ❄ ✲ H B i ❄ ✲ · · · ✲ H B ❄ Pr o of. The pr o of of lemma 4.4.1 shows that the ma p ψ : E k → H B k preserves the inner pro duct. Let p = φψ , so H B k ∼ = (1 − p ) H B k ⊕ p H B k is an inner pro duct preserving cb-iso morphism. Now use the Eilenberg swindle E k ⊕ H B k ∼ = E k ⊕ (1 − p ) H B k ⊕ ( p H B k ⊕ (1 − p ) H B k ⊕ · · · ) ∼ = H B k , to obta in an inner pro duct preserving isomo rphism. Note that th e inﬁnite direct sum is well deﬁned cf.3.3.1 1, since only t wo diﬀerent mo dules app ear in it.  Lemma 4.4 .4. L et B b e a C k -algebr a with sp e ctr al triple ( H , D ) . The algebr as K B k ( H B k ) and E nd ∗ B k ( H B k ) ar e c ompletely *-isomorphic t o close d sub algebr as of Sob k (1 ⊗ D ) of the selfadjoint op er ator 1 ⊗ D in H ˜ ⊗ H . In p articular they ar e sp e ctr al invari ant in their C ∗ -closur es. Pr o of. It is immediate that K B k ( H B k ) ∼ = K ⊗B k is a close d subalgebra of So b k (1 ⊗ D ), s ince the W oronowics pro jections s atisfy p 1 ⊗ D = 1 ⊗ p D . By theorem B.7 this extends to an in volutiv e res presentation o f M ( K B k ( H B k )) ∼ = End ∗ B k ( H B k ).  Lemma 4.4.5. L et B b e a C k -algebr a and E ⇌ B a C ∗ -mo dule. Then E is a C k -mo dule if a nd only if t her e is an appr oximate unit (4.20) u n = X 1 ≤| i |≤ n x i ⊗ x ′ i ∈ Fin B ( E ) , such that kh x ′ i , x j ik k ≤ C . Pr o of. The implicatio n ⇒ is trivial, a s x i = x ′ i in deﬁnition 4.3.1. F or the o ther direction, no te that q = ( h x ′ i , x j i ) is an idemp otent in End ∗ B k ( H B k ), a nd by lemma 4.4.4 the range pro jection p (coro lla ry 4.2 .5) is a n element of End ∗ B k ( H B k ) and p H B k = q H B k . Therefo r e E is a C k -mo dule with approximate unit u n = P 1 ≤| i |≤ n pe i ⊗ pe i , with { e i } the standard basis.  Prop ositi o n 4.4. 6. L et E ⇌ B and F ⇌ C b e C k -mo dules with appr oximate units u n = X 1 ≤| i |≤ n x i ⊗ x i , v n = X 1 ≤| i |≤ n y i ⊗ y i , 30 BRAM M ESLAND r esp e ctively. If π : B k → End ∗ B k ( F k ) is a c ompletely b ounde d h omomorphi sm, then E k ˜ ⊗ B k F ⇌ C is c ompletely i somorphic to a C k -mo dule with ap pr oximate unit u n,m = X 1 ≤| i |≤ n X 1 ≤| j |≤ m ( x i ⊗ y j ) ⊗ ( y j ⊗ x i ) , inner pr o duct h e ⊗ f , e ′ ⊗ f ′ i : = lim n X 1 ≤| i |≤ n hh x i , e i f , h x i , e ′ i f ′ i , (4.21) and C i -submo dules cb-isomorphic to E k ˜ ⊗ B k F i . Pr o of. Note that y j ⊗ x i denotes the functional e ⊗ f 7→ h y j , π ( h x i , e i ) f i . The approximate unit u n,m deﬁnes a stably rig ged structure on E k ˜ ⊗ B k F k . (Note tha t strictly s pe a king, u n,m should b e reindexed to bring it in the form (4.20), a nd that this can b e done without pr oblems). The homomorphism B k → End ∗ C k ( F k ) in particula r gives maps B k → End ∗ B i ( F i ) fo r all i ≤ k , and hence stably r ig ged structures on each E k ˜ ⊗ B k F i . F ro m theorem 3 .3 .9, it follows that E k ˜ ⊗ B k F is completely isomor phic to a C ∗ -mo dule. The inner pro duct corr e sp onding to u n.m is h e ⊗ f , e ′ ⊗ f ′ i : = lim n,m X 1 ≤| i |≤ n X 1 ≤| j |≤ m hh x i , e i f , y j ih y j , h x i , e ′ i f ′ i = lim n X 1 ≤| i |≤ n hh x i , e i f , h x i , e ′ i f ′ i . This shows that the functional y j ⊗ x i do es not coincide with the functiona l deﬁned by x i ⊗ y j via this innerpro duct when π is not a *-homomr phism. How ever, the approximate un it u n,m satisﬁes lemma 4.4.5 , and the C i -submo dules ar e clea r ly cb-isomorphic to E k ˜ ⊗ B k F i .  4.5. R egular ope rators on C k -mo dul es. T o develop the theo ry o f regular op- erators in C k -mo dules, w e need to str engthen deﬁnition 1 .3.1 a little bit, due to the ﬁner top olog y on s uch mo dules. Mor eov er , due to the a bsence of squa re ro ots we have to develop the theory for the selfadjoint case ﬁrst. Deﬁnition 4.5.1. Let B be a C k -algebra and E ⇌ B a C k -mo dule. (1) A closed densely deﬁned selfa djoint op era tor D : Dom D → E k is r e gular if the o pe rators ( D ± i ) − 1 are densely deﬁned and hav e ﬁnite k -nor m. (2) A closed densely deﬁned op erator D : Dom D → E k is r e gular if D ∗ is densely deﬁned a nd the selfadjoint o per ator  0 D ∗ D 0  is reg ular. Note that this deﬁnition in par ticular implies that D ± i have dense range. W e wish to prov e th e analogue of the W oronowicz theorem 1.3.3 for suc h opera tors. Along the wa y w e will ﬁnd cleaner, eq uiv a lent c har a cterizations of regularity , e s - pec ially for selfadjoint op er ators. Howev er, these seem to be harder to verify in practice. Prop ositi o n 4. 5 .2. L et B b e a C k -algebr a and E ⇌ B a C k -mo dule. Supp ose D is a selfadjoint r e gular o p er ator in E k . Then D 2 is densely d eﬁne d, Dom D 2 is a c or e for D , and the op er ators 1 + D 2 : Dom D 2 → E k , BIV ARIANT K -THEOR Y AND CORRESPONDENCES 31 D ± i : Dom D → E k ar e bije ctive. Pr o of. The op erato rs D ± i hav e dense range, and by a ssumption their in verses extend to mutually adjoint elements r + , r − of End ∗ B k ( E k ). Similarly , one g ets ad- joint able extensions of D r + and r − D , whic h ar e adjoint to one another. W e have that Im r + ⊂ Dom D , by taking a sequence e n ∈ Im ( D + i ), such that e n → e . Then r + e n → r + e and D r + e n → D r + e and since D is clo sed, r + e ∈ Dom D . Also r + ( D + i ) e = e a nd hence Im r + = Dom D . The same holds for r − . Now obs e rve that for e ∈ Dom D a nd f ∈ E k h e, f i = h r ± ( D ± i ) e, f i = h e, ( D ∓ i ) r ∓ f i , so f = ( D ± i ) r ± f and D ± i are surjective. This in particular implies that 1 + D 2 is surjective. Let f ∈ Dom D , then f = r + e for some e ∈ E k . Cho ose a se q uence e n in Im (1 + D 2 ) = E k with r − e n → e . Then r + r − e n → f and D r + r − e n → D f . But r + r − e n ∈ Do m D 2 since e n ∈ Im (1 + D 2 ), so Dom D 2 is a core for D and in particular is dense.  Corollary 4.5.3. L et D b e a close d densely deﬁne d op er ator in E k with densely deﬁne d adjoint. Then D is r e gular if and only if 1 + D ∗ D , 1 + D D ∗ ar e surje ctive. If D is r e gular then D ∗ is r e gular and Dom D ∗ D is a c or e for D . Pr o of. Cons ider the se lfadjoint op erator ˜ D =  0 D ∗ D 0  , which is regula r in E k ⊕ E k if a nd o nly if D is regular in E k . ⇒ This is the statement that (1 + ˜ D 2 ) is sur jective. ⇐ Since ( ˜ D + i )( ˜ D − i ) =  1 + D ∗ D 0 0 1 + D D ∗  , this implies that ( ˜ D ± i ) are bijectiev e, and hence by Banac h-Steinhaus their in- verses are bo unded and adjointable, hence D is regular. The other statements are immediate from prop o s ition 4.5.2  Theorem 4 . 5.4. L et B b e a smo oth C ∗ -algebr a and E ⇌ B a smo oth C ∗ -mo dule. Supp ose D is a densely deﬁne d close d o p er ator in E k , with densely deﬁne d adjoint. Then D is r e gular if and only if G ( D ) ⊕ v G ( D ∗ ) ∼ = E k ⊕ E k unitarily. Pr o of. W e may ass ume that D is selfadjoint, using the sa me tric k as in corollary 4.5.3. T he preceeding lemma s show that the o p er ators (1 + D 2 ) − 1 , D (1 + D 2 ) − 1 and D 2 (1 + D 2 ) − 1 are selfadjoint elements of E nd ∗ B k ( E k ). Ther efore w e ca n write down a W o ronowicz pro jection p D :=  (1 + D 2 ) − 1 D (1 + D 2 ) − 1 D (1 + D 2 ) − 1 D 2 (1 + D 2 ) − 1  . 32 BRAM M ESLAND It ma ps E k ⊕ E k int o G ( D ). F rom the relation (1 + D 2 ) − 1 + D 2 (1 + D 2 ) − 1 = 1 it follows that 1 − p D maps E k ⊕ E k int o v G ( D ). Since these submo dules are orthogo nal, their sum m ust b e all of E k ⊕ E k . The converse follo ws by a standard ar g ument as in [2 3]. Let p b e the pro jection onto G ( D ), p =  a b ∗ b d  . Then Im a ⊂ Dom D, b = D a and Im b ⊂ Dom D, 1 − a = D b . Thus, Im a ⊂ Dom D 2 and 1 − a = D 2 a . Then (1 + D 2 ) a = 1 , so (1 + D 2 ) is surjective and D is reg ular.  Corollary 4.5. 5. A densely deﬁne d close d symmetric op er ator in E k is selfadjoint and r e gular if and only if G ( D ) ⊕ v G ( D ) ∼ = E k ⊕ E k . Combining this with pro po sition 4.3.2, we see that regula r op er ators in E k extend to E i ∼ = E k ˜ ⊗ B k B i for all i ≤ k as D ⊗ 1, and this extension pr eserves selfadjointness. Corollary 4.5.6 (cf.[22], lemma 2.3) . If D is a r e gular op er ator in E k such that D and D ∗ have dense r ange, then D − 1 is r e gular and D − 1 ∗ = D ∗− 1 . In p articular, if S, T ∈ End ∗ B k ( E k ) have dense ra nge, and adjoints with dense r ange, then S − 1 T − 1 is r e gular with adjoint T ∗− 1 S ∗− 1 . Pr o of. This follows by observ ing that the unitary v maps the g raph of D to that of − D − 1 .  Theorem 4.5. 7 . L et D b e a densely deﬁne d close d symmetric op er ator in E k . The fol lowing ar e e quivalent: (1) D is self adjoi nt and r e gu lar; (2) The op er ators D ± i : Dom D → E k ar e bije ctive; (3) Im ( D + i ) ∩ Im ( D − i ) is dense and t he op er ators ( D ± i ) − 1 have b ounde d k -norm. Pr o of. 1 . ) ⇒ 2 . ) W e alr eady saw in propositio n 4.5.2 that for s e lfadjoint regular op erator s, D ± i ar e bijective. 2 . ) ⇒ 3 . ) F o llows from theor em 4.4 .2. 3 . ) ⇒ 1 . ) The extensio ns r ± ∈ End ∗ B k ( E k ) of ( D ± i ) − 1 are mut ually adjoint b eca use r ∗ + e = ( D + i ) − 1 ∗ e = ( D − i ) − 1 e = r − e, for e ∈ Im ( D + i ) ∩ Im ( D − i ) and this subset is dense. O ne then shows as in prop osition 4 .5 .2 that in fa ct ( D ± i ) are bijectiv e and th us r ± = ( D ± i ) − 1 . F r om corolla r y 4.5.6 we then g e t that D ± i are regular and mutually adjoint, hence D is selfadjoint and regular .  A regular op erator in E k is almost selfadjoint if it satisﬁes the analog ue of deﬁnition 1.3.6, and the prov es of prop osition 1.3.7 and its corolla r y 1.3.8 g o through verbatim. Tha t is, for an almost selfadjoint op erator and λ ∈ R suﬃciently large, D ± λi a nd D ∗ ± λi are bijections Dom D → E k and the formula p = (1 + D 2 λ 2 ) − 1 D λ 2 (1 + D 2 λ 2 ) − 1 D (1 + D 2 λ 2 ) − 1 D 2 λ 2 (1 + D 2 λ 2 ) − 1 ! , deﬁnes an idemp otent in End ∗ B k ( E k ) with r ange Im p = G ( D ), satisfying v p v ∗ = 1 − p . BIV ARIANT K -THEOR Y AND CORRESPONDENCES 33 4.6. T ransv erse smo o thne ss. Regular op era tors in a C k -mo dule b ehav e simi- larly to those in C ∗ -mo dules. In par ticular, their graphs are complemented sub- mo dules given by W oro nowicz pro jections . This means that for a subalgebra A ⊂ E nd ∗ B k ( E k ), the repr e sentations π n (4.12), θ n (4.13) and χ n (corollar y 4 .1.5) can be deﬁned, rela tive to a regular op era tor D in E k . They have the sa me prop- erties as tho se in a C ∗ -mo dule. Strictly speaking we should denote them b y π k n , θ k n and χ k n , but w e suppress this in the no tation, unles s it causes confusion. In particular, Sob elev algebras Sob k n ( D ) and A n := A ∩ Sob k n ( D ) are deﬁned for a *-subalge br a A ⊂ End ∗ B k ( E k ). W e can construct a Sobolev chain E k j for an almo st selfadjoin t D , and v ie w it as a mor phism of in verse s ystems, just as in corollary 1.3.5 and the prop o s ition preceeding it. Prop ositi o n 4 . 6.1. L et E ⇌ B b e a C k -mo dule and D an almost selfadjoint r e gular op er ator in E k . Then the Sob olev mo dules E n of D ar e C k over B . Pr o of. Let u n = P 1 ≤| i |≤ n x i ⊗ x i be a C k - approximate unit for E . O ne chec ks that the map E k → G ( D ) ⊂ E k ⊕ E k e 7→ p  ie e  , preserves the inner pro duct. Therefore u 1 n := X 1 ≤| j |≤ n p  ix j x j  ⊗ p  ix j x j  , satisﬁes the req uirement of deﬁnition 4 .3.1, and hence smo othens (up to deg ree k ) the ﬁrst So bo lev module E 1 . D 2 , viewed as an operato r in G ( D ) = E 1 is C k for this approximate unit: Since (4.22) h p  ix j x j  ,  e D e  i = h  ix j x j  ,  e D e  i , the mo dules E i 1 ⊂ E i ⊕ E i and G ( D ) i coincide a s submo dules of E ⊕ E , for i ≤ k , and the nota tion E i 1 is unam biguous. T he r efore D 2 restricts to a s e lfa djoint regular op erator in each E i 1 , i ≤ k . Next w e pro ceed by induction. Given that E m is smo oth, we use its app oximate unit u m n and the W oro nowicz pro jection of D n +1 to construct a C k approximate unit u m +1 n for E n +1 = G ( D n +1 ). By 4 .2 2, the mo dules E i n +1 and G ( D n +1 ) i coincide, and since D n +1 is selfadjoint reg ula r in e a ch E i n , so is D n +2 in E i n +1 .  34 BRAM M ESLAND The situation of the pr evious pr op osition can b e visualiz e d in a diag ram o f completely co nt ractive injections: . . . . . . . . . . . . · · · ✲ E j +1 i +1 ❄ ✲ E j +1 i ❄ ✲ E j +1 i − 1 ❄ ✲ · · · ✲ E j +1 ❄ · · · ✲ E j i +1 ❄ ✲ E j i ❄ ✲ E j i − 1 ❄ ✲ · · · ✲ E j ❄ . . . ❄ . . . ❄ . . . ❄ . . . ❄ · · · ✲ E i +1 ❄ ✲ E i ❄ ✲ E i − 1 ❄ ✲ · · · ✲ E . ❄ W e can now deﬁne the notion o f transversely smo oth K K -cycle. Deﬁnition 4. 6.2. Let A, B b e C k -algebra s, E ⇌ B a C k -bimo dule and D a regular op erator in E k . The pair ( E , D ) is tr ansverse C k if the s ubalgebras A i , ar e mapp ed completely b oundedly in to Sob i i ( D ) for a ll i ≤ k . It is a C k -cycle when a ( D ± i ) − 1 ∈ K B i ( E i ) for a ∈ A i , i ≤ k . Note that there are completely bounded injections Sob i i ( D ) → So b i − 1 i ( D ), and Sob i i ( D ) → Sob i i − 1 ( D ) th us that transverse smo o thness implies A k gets ma pped completely b oundedly into Sob i j ( D ) for all i, j ≤ k . 4.7. Bo unded p erturbations. The following characteriza tion of the the do main of the represe ntations π n is in teresting in itself. It is a r elative b oundedness con- dition. A sligh tly w eaker form of this condition turns out to b e suﬃcient for an op erator R ∈ End ∗ B k ( E k ) to belong to the domain of θ n . W e use the notation a d D for the deriv atio n a 7→ [ D, a ]. Prop ositi o n 4.7.1. L et E b e a C k -mo dule over the C k -algebr a B , D a r e gular op er ator in E k , and a ∈ End ∗ B k ( E k ) . (1) a ∈ Sob k n ( D ) if and only if ∀ m ≤ n : (ad( D ) m a )( D ± i ) − m +1 ∈ E nd ∗ B k ( E k ) and ( D ± i ) − m +1 (ad( D ) m a ) ∈ E nd ∗ B k ( E k ) . (2) If ∀ m ≤ n : (ad( D ) m a )( D ± i ) − m ∈ End ∗ B k ( E k ) and ( D ± i ) − m (ad( D ) m a ) ∈ End ∗ B k ( E k ) , then [ D , θ n − 1 ( a )]( D ± i ) − 1 , ( D ± i ) − 1 [ D , θ n − 1 ( a )] ar e adjointable and henc e θ n ( a ) ∈ E nd ∗ B k ( G ( D n ) ⊕ v [ n ] G ( D n )) . That is, a ∈ Dom θ n . BIV ARIANT K -THEOR Y AND CORRESPONDENCES 35 Pr o of. W e only pr ov e (1 ), as (2) can b e done by the same metho d. ⇒ F o r n = 1 the statement reduces to the b oundednes s of the commutators [ D , a ]. Pro ceeding by induction, we assume the statement prov e n for m ≤ n . Let a ∈ A n +1 , i.e. a ∈ A n and [ D, θ n ( a )] ∈ End ∗ B k ( L 2 n j =1 E k ). W e prove that ad( D ) n +1 a ( D ± i ) − n ∈ E nd ∗ B k ( E k ). Since θ n ( a ) is an orthogo nal sum θ n ( a ) = p n θ n ( a ) p n + p ⊥ n θ n ( a ) p ⊥ n = θ n ( a ) p n + p ⊥ n θ n ( a ) , [ D , θ n ( a ) p n ] a nd [ D , p ⊥ n θ n ( a )] are b oth b ounded. Now since θ n ( a ) p n = π n ( a ) p n − 1 p n =  θ n − 1 ( a ) 0 [ D , θ n − 1 ( a )] θ n − 1 ( a )  p n − 1 p n , [ D , θ n ( a ) p n ] is b ounded if and only if [ D , [ D , θ n − 1 p n − 1 ]](1 + D 2 ) − 1 , and [ D , [ D , θ n − 1 p n − 1 ]] D (1 + D 2 ) − 1 , are adjointable. This in turn is true if and only if [ D , [ D , θ n − 1 ( a ) p n − 1 ]]( D ± i ) − 1 is adjointable. The s ame arg ument c a n now b e applied another n − 1 -times to yield that (ad D ) n +1 ( a )( D ± i ) − n ∈ End ∗ B k ( E k ) . One proves that ( D ± i ) − k +1 ad( D ) k a ∈ End ∗ B i ( E i ) , in the same way by using the summa nd p ⊥ n θ n ( a ) p ⊥ n . ⇐ Suposse that a ∈ A n . Then θ n ( a ) is adjointable. The ab ove metho d shows that [ D , θ n ( a )] is adjointable whenever [ D , θ n − 1 ( a )] a nd [ D , [ D , θ n − 1 ( a )]]( D ± i ) − 1 are adjointable. As a b ov e, this arg ument can be repea ted to ﬁnd that ∀ k ≤ n : ad( D ) k a ( D ± i ) − k +1 ∈ End ∗ B k ( E k ) a nd ( D ± i ) − k +1 ad( D ) k a ∈ End ∗ B k ( E k ).  Bounded per turbations by ele men ts in Dom θ n are w ell b ehaved with r e sp ect to taking Sobolev chains up to degree n + 1 . Eq uiv a lently , such pe r turbations preserve the do main op the p owers of D up to degree n + 1. Deﬁnition 4.7.2. Let E , F b e C k -mo dules over a C k -algebra B . A t op olo gic al isomorphi sm of C k -mo dules is an in vertible ele men t g ∈ Hom ∗ B k ( E k , F k ). Obviously , a top ologica l iso morphism of C k -mo dules induces topo lo gical isomor- phisms of C i -mo dules for i ≤ k . Lemma 4. 7.3. L et D b e a selfadjo int r e gu lar op er ator in the C k -mo dule E and let R ∈ E nd ∗ B k ( E k ) . The map g : G ( D ) → G ( D + R ) ( e, ( D + R ) e ) 7→ ( e , D e ) , is a t op olo gic al isomorphism of C k -mo dules. Pr o of. On E k ⊕ E k , the map g can b e written as g = p D  1 0 − R 1  p D + R , 36 BRAM M ESLAND and hence it is an adjointable o p e rator. Its in verse is g − 1 = p D + R  1 0 R 1  p D .  When R preserves the domain of D m for all m ≤ n , w e can inductiv ely deﬁne maps g m : G (( D + R ) m ) → G ( D m ) for m ≤ n + 1, b y setting g m +1 ( e, ( D + R ) e ) := ( g m ( e ) , D g m ( e )) . Theorem 4.7.4. If R ∈ Do m θ m , for a l l m ≤ n , then the c anonic al map s g k : G (( D + R ) m ) → G ( D m ) , ar e top olo gic al isomorphi sms of C k -mo dules for al l m ≤ n + 1 . Pr o of. F or n = 0 the a b ove lemma applies. Pr o ceeding by induction, w e supp ose the theorem proven fo r n − 1. The hypothesis imply that θ n − 1 ( R ) ∈ End ∗ B i ( G ( D n − 1 ) ⊕ v [ n − 1] G ( D n − 1 )), and hence χ n − 1 ( R ) ∈ E nd ∗ B k ( G ( D n − 1 )). The induction hypo thesis gives iso mo rphisms g n ⊕ g n : G (( D + R ) n ) ⊕ G (( D + R ) n ) → G ( D n ) ⊕ G ( D n ) , under which the gra ph G (( D + R ) n +1 ) → G ( D n +1 + χ n ( R )) , bijectiv ely . This can b e seen by inductio n: F or m = 1, the cla im is o bvious. Suppo se that g m ⊕ g m maps G (( D + R ) m +1 ) bijectiv ely to G ( D m +1 + χ m ( R )). This is equiv a le nt to saying tha t g m ( D + R ) e = Dg m ( e ) + χ m ( R ) g m ( e ) . Then g m (( D + R ) e, ( D + R ) 2 e ) = ( D g m e + χ m ( R ) g m ( e ) , D 2 g m e + D χ m ( R ) g m e ) , and since ( χ m ( R ) g m e, D χ m ( R ) g m e ) = χ m +1 ( R )( g m e, D g m e ), g m +1 ⊕ g m +1 is a bijection G (( D + R ) m +2 ) → G ( D m +2 + χ m +1 ( R )) , and the claim fo llows. Since w e hav e χ n ( R ) ∈ E nd ∗ B k ( G ( D n )) , by lemma 4.7.3, the map G ( D n +1 + χ n ( R )) → G ( D n +1 ) ( e, D n e + χ n ( R ) e ) 7→ ( e, D n +1 e ) , is a to po logical isomorphism, and the comp osition of these t wo maps restr icted to G (( D + R ) n +1 ) is the canonical map g n +1 .  Corollary 4.7.5. If R ∈ Sob k n ( D ) , or i f it satisﬁes (2) of pr op osition 4.7.1, the c anonic al ma ps g m : G (( D + R ) m ) → G ( D m ) , ar e top olo gic al isomorphi sms of C k -mo dules for al l m ≤ n + 1 . BIV ARIANT K -THEOR Y AND CORRESPONDENCES 37 Pr o of. Both conditions imply R ∈ Dom θ n , s o the theorem applies.  Corollary 4.7.6. If R ∈ Sob k n ( D ) , then Sob k m ( D ) = Sob k m ( D + R ) for al l m ≤ n + 1 . Pr o of. The statement clear ly ho lds for n = 0. Supp ose it holds fo r n − 1 and let R = R ∗ ∈ So b k n ( D ). By the induction hypothesis , Sob k n ( D ) = Sob k n ( D + R ). The isomorphism g n : G (( D + R ) n ) → G ( D n ) intertwines the representations χ D + R n and χ D n , a nd for a ∈ Sob k n ( D ) = Sob k n ( D + R ) w e hav e g n [ D + R, χ D + R n ( a )] g − 1 n = [ D + χ D n ( R ) , χ D n ( a )] . Thu s, cf. cor ollary 4.1 .6 a ∈ So b k n +1 ( D ) if and only if a ∈ Sob k n +1 ( D + R ).  5. Connections Connections on Riemannian manifolds ar e a vital to ol for diﬀerentiating func- tions and v ector ﬁelds over the manifold. Cunt z and Q uillen [15] develop ed a pur ely algebraic theory of connections on mo dules, whic h gives a b eautiful characteriza- tion of pro jective mo dules. They a re exactly those mo dules that admit a universal connection. W e revie w their results, but will rec ast everything in the setting o f op- erator modules. This is only straig ht forward, b ecause the Haag e r up tensor pr o duct linearizes the m ultiplication in an o p erator algebra in a con tinuous wa y . W e then pro ceed to co nstruct a categ ory o f mo dules with connection, and ﬁnally pa ss to inv erse s ystems o f mo dules. 5.1. Universal forms. The notion of universal diﬀerential fo r m is widely used in noncommutativ e geometr y , esp ecially in connectio n with cyclic homology [1 2]. F o r top ological alg ebras, their exac t deﬁnition dep ends on a choice of topolo gical tensor pro duct. The default choice is the Grothendieck pro jective tensor pr o duct, beca use it linearizes the m ultiplication in a top ologica l algebr a contin uously . How ever, when dealing with o pe r ator algebr as, the natura l choice is the Haager up tensor pro duct. Meyer [25] has shown any op era tor algebra B admits a canonical unit ization B + , simply by ta king the unital algebra genera ted by it in any completely isomorphic representation π : B → B ( H ), on some Hilbert space H . Note that, when B ha s a unit, then q = π (1) is an idempo tent , s o π restricts to a unita l representation on B ( q H ) and the unitization coincides with B in this case. In this section we will alwa ys repla ce B b y B + . Deﬁnition 5.1.1. Let B be a n op erator a lg ebra. The mo dule of un iversal 1 -forms ov er B is deﬁned as Ω 1 ( B ) := ker ( m : B ˜ ⊗B → B ) . Here m is the gr aded m ultiplicatio n map m ( a ⊗ b ) = ( − 1) ∂ b ab . By deﬁnition, there is an exact seq uence of op era to r bimo dules 0 → Ω 1 ( B ) → B ˜ ⊗B m − → B → 0 . The mo dule Ω 1 ( B ) inherits a grading from B ˜ ⊗B . The map d : B → Ω 1 ( B ) a 7→ 1 ⊗ a − ( − 1) ∂ a a ⊗ 1 is an even graded bimo dule deriv a tion. 38 BRAM M ESLAND Lemma 5.1.2. The derivation d is universal. F or a ny c ompletely b ounde d gr ade d derivation δ : B → M int o an B op er ator bimo dule, ther e is a unique c ompletely b ounde d bimo dule homomorph ism j δ : Ω 1 ( B ) → M such tha t the di agr am B δ ✲ M Ω 1 ( B ) j δ ✲ d ✲ c ommutes. If δ is homo gene ous, t hen s o j δ and ∂ δ = ∂ j δ . Pr o of. Set j δ ( da ) = δ ( a ). This determines j δ bec ause da generates Ω 1 ( B ) a s a bimo dule.  An y deriv a tion δ : B → M has its asso cia ted mo dule o f fo rms Ω 1 δ := j δ (Ω 1 ( B )) ⊂ M . Deﬁnition 5.1.3. Let δ : B → M b e a g raded de r iv ation as a b ov e, and E a right op erator B -mo dule. A δ - c onne ction on E is a completely b ounded even linear ma p ∇ δ : E → E ˜ ⊗ B Ω 1 δ , satifying the Leibniz rule ∇ δ ( eb ) = ∇ δ ( e ) b + e ⊗ δ ( b ) . If δ = d , the connectio n will be denoted ∇ , a nd r eferred to as a un iversal c onne ction . Note that a univ e r sal connection ∇ on a mo dule E gives rise to δ - connections for any completely b ounded deriv ation δ , simply by setting ∇ δ := 1 ⊗ j δ ◦ ∇ . If δ is of the form δ ( a ) = [ S, a ], for S ∈ Hom c C ( X, Y ), where X a nd Y ar e left A -op era tor mo dules, we wr ite simply ∇ S for ∇ δ . Not all mo dules admit a univ ers al connection. Cuntz and Quillen show ed that universal connections c hara cterize alg ebraic pro jectivit y . Their proof sho ws that stably rigged modules admit universal connections, but the class of modules admit- ting a connection might b e large r . Prop ositi o n 5.1.4 ([15]) . A right B op er ator mo dule E admits a universal c on- ne ction if a nd only if the multiplic ation map m : E ˜ ⊗B → E i s B -split. Pr o of. Cons ider the ex a ct sequence 0 ✲ E ˜ ⊗ B Ω 1 ( B ) j ✲ E ˜ ⊗B m ✲ E ✲ 0 , where m is the multiplication map and j ( e ⊗ da ) = e b ⊗ 1 − e ⊗ b . A linear ma p s : E → E ˜ ⊗B determines a linear map ∇ : E → E ˜ ⊗ B Ω 1 ( B ) by the formula s ( e ) = e ⊗ 1 − j ( ∇ ( e )), since j is injectiv e. Now s ( eb ) − s ( e ) b = j ( ∇ ( e ) b + e ⊗ db − ∇ ( eb )) , whence s b eing a n B -mo dule map is equiv a lent to ∇ being a connection.  BIV ARIANT K -THEOR Y AND CORRESPONDENCES 39 Corollary 5.1. 5 . Any st ably rigge d mo dule E over B admits a c onne ction. Pr o of. E is a direct summand in H B , i.e. E = p H B , with p 2 = p ∈ End ∗ B ( H B ). Observe tha t H B ˜ ⊗ B Ω 1 B ∼ = H ˜ ⊗ Ω 1 ( B ). Consider the Gr assmann c onne ction d : H B → H ˜ ⊗ Ω 1 ( B ) h ⊗ a 7→ h ⊗ da, and deﬁne p ∇ p : E → E ˜ ⊗ B Ω 1 ( B ).  5.2. Pro duct connections. W e now proceed to connections on tensor pro ducts of stably rigged mo dules. An ticipating the use of connections o n un b ounded bi- mo dules, a categor y of modules with connection is constr ucted. Prop ositi o n 5.2.1. L et E b e a st ably rigge d B -mo dule with a u n iversal c onne ction ∇ , F a stably r igge d ( B , C ) -bimo dule with universal c onne ction ∇ ′ . Then ∇ and ∇ ′ determine a u niversal C -c onne ction 1 ⊗ ∇ ∇ ′ : E ˜ ⊗ B F → E ˜ ⊗ B F ˜ ⊗ C Ω 1 ( C ) . Pr o of. Cons ider the der iv ation δ : B → End C ( F, F ˜ ⊗ C Ω 1 ( C )) b 7→ [ ∇ ′ , b ] . By universality there is a unique map j δ : Ω 1 ( B ) → Ω 1 δ , int ertwining d and δ . Thus, ∇ induces a connection ∇ δ : E → E ˜ ⊗ B Ω 1 δ , by comp os ing with j δ . Subsequently deﬁne 1 ˜ ⊗ ∇ ∇ ′ : E ˜ ⊗ B F → E ˜ ⊗ B F ˜ ⊗ C Ω 1 ( C ) e ⊗ f 7→ e ⊗ ∇ ′ ( f ) + ∇ δ ( e ) f , which is a connection.  W e w ill refer to the connection of prop ositio n 5.2 .1 as the pr o duct c onn e ction . T a king pro duct connections is in fact a sp ecial case of the following construction. Let ( E , ∇ ) be a graded stably rigged righ t B mo dule with c o nnection and D ∈ Hom c ( X, Y ) a homog eneous op era tor b etw een graded left B op erator mo dules X , Y . Denote by 1 ˜ ⊗ ∇ D the op erato r (5.23) 1 ⊗ ∇ D ( e ⊗ x ) := ( − 1) ∂ D ∂ e ( e ⊗ D ( x ) + ∇ D ( e ) x ) , which is a w ell deﬁned op era tor E ˜ ⊗ B X → E ˜ ⊗ B Y . This construction is as s o ciative up to isomorphis m. Theorem 5 .2.2. Le t E b e a stably rigge d B -mo dule, F a st ably r igge d ( B , C ) - bimo dule and ∇ , ∇ ′ universal c onne ctions on E and F r esp e ctively. F urthermor e let X , Y b e left op er ator C - mo dules, and D ∈ Hom c ( X, Y ) . Then 1 ˜ ⊗ ∇ 1 ˜ ⊗ ∇ ′ D = 1 ˜ ⊗ 1 ˜ ⊗ ∇ ∇ ′ D , under the intert wining isomorphism. 40 BRAM M ESLAND Pr o of. Since ∂ D = ∂ 1 ⊗ ∇ D a nd ( − 1) ∂ ( e ⊗ f ) ∂ D = ( − 1) ( ∂ e + ∂ f ) ∂ D = ( − 1) ∂ e∂ 1 ⊗ ∇ D ( − 1) ∂ f ∂ D , we assume that D is even, the o dd case diﬀering only by this factor . Reca ll the formula for the pr o duct connection 1 ⊗ ∇ ∇ ′ ( e ⊗ f ) := e ⊗ ∇ ′ ( f ) + ∇ δ ( e ) f . Moreov er, wr ite ∇ D for ∇ ∇ ′ D . It is straig htforward to check that (1 ˜ ⊗ ∇ ∇ ′ ) D ( e ⊗ f ) = e ⊗ ∇ ′ D ( f ) + ∇ D ( e ) f . Therefore we hav e 1 ⊗ 1 ⊗ ∇ ∇ ′ D ( e ⊗ f ⊗ x ) = e ⊗ f ⊗ D x + 1 ⊗ ∇ ∇ ′ ( e ⊗ f ) x = e ⊗ f ⊗ D x + e ⊗ ∇ ′ D ( f ) x + ∇ D ( e )( f ⊗ x ) . On the other hand 1 ˜ ⊗ ∇ 1 ˜ ⊗ ∇ ′ D ( e ⊗ f ⊗ x ) = e ⊗ (1 ˜ ⊗ ∇ ′ D )( f ⊗ x ) + ∇ 1 ˜ ⊗ ∇ ′ D ( e )( f ⊗ x ) = e ⊗ f ⊗ Dx + e ⊗ ∇ ′ D ( f ) x + ∇ 1 ˜ ⊗ ∇ ′ D ( e )( f ⊗ x ) , th us, it suﬃces to show that ∇ D = ∇ 1 ˜ ⊗ ∇ ′ D . T o this end, obser ve that [1 ˜ ⊗ ∇ ′ D , b ] = [ ∇ ′ D , b ] : E ⊗ B F → E ⊗ B F, which gives a natural isomorphism Ω 1 ∇ ′ D ∼ − → Ω 1 1 ˜ ⊗ ∇ ′ D int ertwining the deriv ations. By universality this gives a commutativ e diagra m Ω 1 ( B ) Ω 1 1 ˜ ⊗ ∇ ′ D ∼ ✲ ✛ Ω 1 ∇ ′ D , ✲ which shows that ∇ D = ∇ 1 ˜ ⊗ ∇ ′ D .  Corollary 5.2.3. L et E , F , G b e s t ably rigge d A -, ( A , B ) -, ( B , C ) -(bi)mo dules, with universal c onne ctions ∇ , ∇ ′ , ∇ ′′ r esp e ctively. The natur al isomorphism E ˜ ⊗ A ( F ˜ ⊗ B G ) ∼ − → ( E ˜ ⊗ A F ) ˜ ⊗ B G intertwines the pr o duct c onne ctions 1 ⊗ 1 ⊗ ∇ ∇ ′ ∇ ′′ and 1 ⊗ ∇ (1 ⊗ ∇ ′ ∇ ′′ ) . The upshot of theore m 5.2.2 and its corolla ry 5.2.3 is that there is a categor y whose ob jects are op erator alg ebras, and whose morphisms Mor( A , B ) are given b y pairs ( E , ∇ ) consisting o f a stably rigg e d right ( A , B )-bimo dule E with a universal B co nnection. The iden tit y morphisms ar e the pair s ( A , d ) c o nsisiting of the trivial bimo dule A and the univ er sal deriv a tion d : A → Ω 1 ( A ). Of course this categ o ry is describ ed equiv alently as the ca tegory o f pairs ( E , s ) o f bimo dules toge ther with a splitting s of the univ er sal exact sequence. One can pro ceed to enrich the ca t- egory describ ed ab ove by considering triples ( E , T , ∇ ) consisting of stably rig ged bimo dules with connection and a distinguished e ndo morphism T ∈ End ∗ B ( E ) . The comp osition la w then b e comes ( E , S, ∇ ) ◦ ( F, T , ∇ ′ ) := ( E ˜ ⊗ B F, S ˜ ⊗ 1 + 1 ˜ ⊗ ∇ T , 1 ˜ ⊗ B ∇ ′ ) . BIV ARIANT K -THEOR Y AND CORRESPONDENCES 41 5.3. Sm o oth connections . W e now consider C k -mo dules over a C k -algebra B . If B is an involutiv e ope r ator algebr a, Ω 1 ( B ) ca rries a natural inv olution, deﬁned by (5.24) ( adb ) ∗ := − ( − 1) ∂ b ( db ∗ ) a ∗ . The inner pro duct on E k induces a pairing E k × E k ˜ ⊗ B k Ω 1 ( B k ) → Ω 1 ( B k ) ( e 1 , e 2 ⊗ ω ) 7→ h e 1 , e 2 i ω . By abus e of notation we write h e 1 , e 2 ⊗ ω i for this pairing. A pairing E k ˜ ⊗ B k Ω 1 ( B k ) × E k → Ω 1 ( B k ) , is obtained by setting h e 1 ⊗ ω , e 2 i := h e 2 , e 1 ⊗ ω i ∗ . A connection ∇ : E k → E k ˜ ⊗ B k Ω 1 ( B k ) , is a *- c onne ction if there is a connection ∇ ∗ on E k for which (5.25) h e 1 , ∇ ( e 2 ) i − h∇ ∗ ( e 1 ) , e 2 i = d h e 1 , e 2 i . The connectio n is Hermitian if we can c ho ose ∇ ∗ = ∇ in the a bove equatio n. Lemma 5.3.1. L et ∇ b e a *-c onn e ction on a C k -mo dule E k . Then ∇ ∗ is unique, and ∇ ∗∗ = ∇ . Pr o of. Let ˜ ∇ be a connection satisfying 5.2 5. By stabilizing and replacing ∇ , ∇ ∗ and ˜ ∇ by ∇ ⊕ d , ∇ ∗ ⊕ d and ˜ ∇ ⊕ d , we ma y assume E k = H B k . F or any c o nnection we have ∇ ( e ) = X i ∈ Z \{ 0 } e i h e i , ∇ ( e ) i , where { e i } i ∈ Z \{ 0 } is the standard basis of H B k . Therefore it suﬃces t o show that h e i , ∇ ∗ ( e ) i = h e i , ˜ ∇ ( e ) i . This fo llows immedia tely from (the adjoint of ) 5.25.  W e now address smo o thness a nd t ransversality of connections on smo o th C ∗ - mo dules. Lemma 5. 3.2. L et E ⇌ B b e a C k -mo dule over a C k -algebr a B , and ∇ : E k → E k ˜ ⊗ B k Ω 1 ( B k ) a *-c onne ct ion. Then ∇ u niquely extends to a *-c onn e ction on E i for al l i ≤ k . Pr o of. Recall the ide ntiﬁcation E i = E k ˜ ⊗ B k B i from pr op osition 4.3 .2, and observe that there is a ca nonical isomo rphism B i ˜ ⊗ B k Ω 1 ( B k ) ˜ ⊗ B k B i → Ω 1 ( B i ) a ⊗ db ⊗ c 7→ a ( db ) c, compatible with d . This allows us to deﬁne ∇ ( e ⊗ b ) := ∇ ( e ) ⊗ b + e ⊗ db , which is easily chec ked to b e a *- c onnection. Uniquenes s follows from the fact that E k is dense in E i for k ≥ i .  The oper ator spaces Sob D n ( E k , E k ˜ ⊗ B k Ω 1 ( B k )) a re deﬁned via r epresentations π n (4.12) and θ n (4.13) on Hom c C ( E k , E k ˜ ⊗ B k Ω 1 ( B k )), relative to a regular op er a tor in E k , using the unbounded op era tor D ⊗ 1 o n E k ˜ ⊗ Ω 1 ( B k ), to make sense of the commutators [ D , · ]. This allows us to spea k of transverse C k -connections. 42 BRAM M ESLAND Deﬁnition 5. 3.3. Let E ⇌ B b e a C k -mo dule ov e r a C k -algebra B , D an almost selfadjoint regular o pe r ator in E k and ∇ a *- connection in E k . Then ∇ is sa id to be tr ansverse C n if ∇ ∈ Sob D n ( E k , E k ˜ ⊗ B k Ω 1 ( B k )). Note that in this deﬁnition, n and k are indep endent of one another. This def - inition can be phrased equiv a lently by sa ying that ∇ ∈ Dom π n − 1 and [ D, θ n ( ∇ )] extends to a completely b o unded op erator E k → E k ˜ ⊗ B k Ω 1 ( B k ). Y et another equiv- alent way of phrasing this (cf. pro p osition 4.7.1) is to say that the op er ators ad( D ) n ( ∇ )( D ± i ) − n +1 , and ( D ± i ) − n +1 ad( D ) n ( ∇ ) , extend to co mpletely bo unded op erator s E k → E k ˜ ⊗ B k Ω 1 ( B k ). It is clear fro m this deﬁnition, that a transverse C n -connection induces a co nnection on the Sob olev chain o f D up to deg ree n . 5.4. Induced op e rators and their graphs. As we ha ve seen, a *-connectio n ∇ : E k → E k ˜ ⊗ B k Ω 1 ( B k ) can b e used to transfer op er ators on F k to E k ⊗ B k F k . W e no w show that th is algebraic pro cedur e is w ell behav ed for selfadjoint regula r op erator s T in F k , and describ e the Sobolev chain, i.e. the gra phs G (1 ˜ ⊗ ∇ T ) j ⊂ E k ˜ ⊗ B k F k ⊕ E k ˜ ⊗ B k F k as top olog ical C k -mo dules, in t erms of the graph of T j , as well as the graph repres entations χ k j (corollar y 4.1 .5). Note that b y prop os ition 4.3.2 E i ˜ ⊗ B i F j ∼ = E j ˜ ⊗ B j F j , whenever i ≥ j . If E carries a C k - left mo dule str ucture from another smoo th C ∗ -algebra A , then E ˜ ⊗ B F carr ie s a canonical C k left A - mo dule structure . Theorem 5 .4.1. Le t k ≥ 1 , A, B , C, C k -algebr as, E , F a C k - ( A, B ) , ( B , C ) - bimo dules, r esp e ct ively. L et T : Dom ( T ) → F k b e selfadjoint and r e gular and tr ansverse C k in F k . If ∇ : E k → E k ˜ ⊗ B k Ω 1 ( B k ) is a *- c onne ct ion, then the op er- ator t := 1 ˜ ⊗ ∇ T is almost sel fadjoi nt and r e gular in E k ˜ ⊗ B k F k . If ∇ is Hermitian, then 1 ˜ ⊗ ∇ T is selfadjoi nt. With g χ 0 = id E ˜ ⊗ B F , for j ≤ i ≤ k , the inductively deﬁne d map g χ j : E i ˜ ⊗ B i G ( T j ) i → G ( t j ) i e ⊗ ( f , T f ) 7→ ( g χ j − 1 ( e ⊗ f ) , 1 ˜ ⊗ ∇ T j ( g χ j − 1 ( e ⊗ f ))) is a top olo gic al isomorphism of C i -mo dules. Mor e over, we have g χ j ◦ ( a ⊗ 1) = χ t j ( a ) g χ i , thus A j → Sob i j (1 ⊗ ∇ T ) c ompletely b oun de d ly, for j ≤ i ≤ k . So in p articular ( E ˜ ⊗ B F , 1 ⊗ ∇ T ) is tr ansverse C k . Pr o of. T o s ee that t j := 1 ⊗ ∇ T j is selfadjoint regular , stabilize E , and denote by d the Grassmannia n connection on H B . Then, via the stabilization isomor phism ∇ ′ := ∇ ⊕ d deﬁnes a C k -*-connectio n on H B ∼ = E ⊕ H B . Since the diﬀerence R := ∇ ′ T − d T is an element of End ∗ C k ( E k ˜ ⊗ B k F k ), it s uﬃce s to prove regula rity o f t w he n ∇ is the Grassmannian connection d o n H B . In that c ase, deﬁne t j on the domain Im (1 ⊗ d ( T j ± i ) − 1 ), which is dense. Then, for e = P m ∈ Z \ 0 e m ⊗ b m , t j : H B i ˜ ⊗ B i G ( T j − 1 ) i → H B i ˜ ⊗ B i G ( T j − 1 ) i e ⊗ f 7→ X m ∈ Z \ 0 e m ⊗ T ( b m f ) . BIV ARIANT K -THEOR Y AND CORRESPONDENCES 43 F o r j = 1 this is symmetric by a standard arg umen t. F or j > 1, B j is repres e n ted on G ( T j − 1 ) by a non ∗ -homomorphism. But then, cf. 4.21 h t j ( e ⊗ f ) , e ′ ⊗ f ′ i = X n ∈ Z \{ 0 } h e n ⊗ T j b n f , e ′ ⊗ f ′ i = X n ∈ Z \{ 0 } X m ∈ Z \{ 0 } hh e m , e n i T j b n f , h e m , e ′ i f ′ i = X n ∈ Z \{ 0 } h T j b n f , b ′ n f ′ i = X n ∈ Z \{ 0 } h b n f , T j b ′ n f ′ i = X n ∈ Z \{ 0 } X m ∈ Z \{ 0 } hh e m , e i f , h e m , e n i T j b ′ n f ′ i = h e ⊗ f , t j ( e ′ ⊗ f ′ ) i . F ur thermore it is closed, selfadjoin t a nd regular, b ecause t j ± i are surjective b y construction: ( t j ± i )(1 ⊗ d ( T j ± i ) − 1 ) = 1 for the connectio n d . F o r the statement on the top olo gical type of G ( t j ) , it again suﬃces to consider the Gr assmannian co nnection on H B k : according to theorem 4.7.4, w e hav e g j : G (( t + R ) j ) ∼ − → G ( t j ) ( x, ( t + R ) x ) 7→ ( g j − 1 x, tg j − 1 x ) C k -top ologica lly , once we show that R = 1 ⊗ d T − 1 ⊗ ∇ T = d T − ∇ T ∈ Do m θ k j . T o this end w e compute (ad(1 ⊗ d T )) j ( R )(1 ⊗ d T ± i ) − j e m ⊗ f = (ad(1 ⊗ d T )) j ( ∇ T − d T ) e m ⊗ ( T ± i ) − j f = X n ∈ Z \{ 0 } (ad(1 ⊗ d T )) j e n ⊗ ω m n ( T ± i ) − j f = X n ∈ Z \{ 0 } e n ⊗ (ad T ) j ( ω m n )( T ± i ) − j f which is an elemen t of End ∗ C k ( E k ⊗ B k F k ) , since the connection is C k . Here the ω m n ∈ Ω 1 T are such that ∇ T ( e m ) = X n ∈ Z \{ 0 } e n ⊗ ω m n . Viewing G ( T i ) a s a submo dule of G ( T i − 1 ) ⊕ G ( T i − 1 ), the repre sentations χ i j : B j → G ( T j ) i , from co rollary 4.1 .5 hav e the form χ i j ( b )( f , T f ) = ( χ i j − 1 ( b ) f , T χ i j − 1 ( b ) f ) , by trans versality .F or conv enience , we suppr ess the χ i j in the notation. By (4 .21), the inner product (inducing an equiv a lent op er a tor space str uc tur e) on H B i ˜ ⊗ B i G ( T i ) 44 BRAM M ESLAND is thus given by h e ⊗ ( f , T f ) , e ′ ⊗ ( f ′ , T f ′ ) i : = X n ∈ Z \{ 0 } hh e n , e i ( f , T f ) , h e n , e ′ i ( f ′ , T f ) i = X n ∈ Z \{ 0 } h ( b n f , T b n f ) , ( b ′ n f ′ , T b ′ n f ′ ) i = X n ∈ Z \{ 0 } h b n f , b ′ n f ′ i + h T b n f , T b ′ n f ′ i . Therefore the map H B i ˜ ⊗ B i G ( T j ) i → G ( t j ) i e ⊗ ( f , T f ) 7→ ( e ⊗ f , t ( e ⊗ f )) , is unitary . F r om this it follows that the A i → Sob i j (1 ⊗ ∇ T ). F or j = 1 this holds b ecaus e [1 ⊗ ∇ T , a ] = [ ∇ T , a ] ⊗ 1 , which is a completely b o unded deriv ation from A 1 int o End ∗ C ( E 1 ˜ ⊗ B 1 F 1 ). Therefore, π 1 ⊗ ∇ T 1 : A 1 → M 2 (End ∗ C ( E 1 ˜ ⊗ B 1 F 1 )) , is completely bounded. Supp os e we ha ve prov en A i → Sob i j (1 ⊗ ∇ T ) completely bo undedly . The isomo rphism g j int ertwines the A i representations, i.e., g j is a bimo dule ma p. F or a ∈ A i +1 , [ t j +1 , χ i +1 j ( a )] = g j ([ ∇ T j +1 , a ] ⊗ 1) , which is a djointable, and the same holds fo r a ∗ . Thus by corolla ry 4 .1.6 [ t j +1 , θ i +1 j ( a )] is adjo intable. It follo ws that a 7→ [ t j +1 , θ i +1 j ( a )] is a completely bounded deriv a- tion A i +1 → M 2 ( G ( t j ). Th usfor j ≤ i , A i +1 → Sob i +1 j +1 (1 ⊗ ∇ T ) completely b ound- edly .  Corollary 5.4. 2. L et k ≥ 1 , A, B , C b e C k -algebr as, ( E , S ) and ( F , T ) t r ansverse C k -bimo dules for ( A, B ) and ( B , C ) r esp e ctively, and ∇ a Hermitian C k -c onne ction on E . If ∇ i s tr ansverse C i , t he op er ators 1 ˜ ⊗ ∇ i T ar e almost sel fadjoi nt, r e gular and tr ansverse C k in G ( S i ) ˜ ⊗ B F , with Sob olev m o dules G ((1 ⊗ ∇ i T ) j ) t op olo gic al ly isomorphi c to G ( S i ) j ˜ ⊗ B j G ( T j ) . Pr o of. The connections ∇ i : E i i → E i i ˜ ⊗ B i Ω 1 ( B i ) are C k *-connections, so the state- men t follows from the pr e v ious theo r em.  5.5. E n do morphism algebras. W e ar e now able to show that the notions of left- and rig ht -smo othness of a C ∗ -mo dule ca n b e treated on equal fo o ting. The no tion of connectio n links the the tw o concepts in an elega n t wa y . Recall the repres entations θ n : A n → End ∗ ( G ( D n )) ⊕ End ∗ ( v [ n ] G ( D n )) , π n +1 : A n +1 → End ∗ ( G ( D n ) ⊕ G ( D n )) . The θ n are endomorphisms of X D n := G ( D n ) ⊕ v [ n ] G ( D n ), resp ecting the direct sum decomp os ition. T he π n act o n X D n ⊕ X D n , but do not respect the direct sum decomp osition. BIV ARIANT K -THEOR Y AND CORRESPONDENCES 45 Theorem 5.5.1. L et E ⇌ B b e a C k -mo dule, ∇ : E k → E k ˜ ⊗ B k Ω 1 ( B k ) a *- c onne ction and F a tr ansverse C k ( B , C ) -bimo dule. Ther e ar e c anonic al t op olo gic al isomorphi sms g π k : E k ˜ ⊗ π k ( X k − 1 ⊕ X k − 1 ) → E k ˜ ⊗ θ k − 1 X k − 1 ⊕ E k ˜ ⊗ θ k − 1 X k − 1 , e ⊗  x y  7→  e ⊗ x e ⊗ y + ∇ T ( e ) x  , g θ k : E k ˜ ⊗ θ k X k → G ( t k ) ⊕ v [ k ] G ( t k ) , e ⊗  x T x  ⊕  − T y y  7→  g θ k − 1 ( e ⊗ x ) t k g θ k − 1 ( e ⊗ x )  + v [ k ] p t k v ∗ [ k ]  − g θ k − 1 ( e ⊗ T y ) g θ k − 1 ( e ⊗ y + ∇ T ( e ) T y )  , wher e t i = (1 ⊗ ∇ T ) i . Mor e over, if E is a tr ansverse C k ( A, B ) -bimo dule, then we have g π k ◦ ( a ⊗ 1) = π t k ( a ) g π k , g θ k ◦ ( a ⊗ 1) = θ t k ( a ) g θ k , that is they ar e bi mo dule map s for the r esp e ctive A k mo dule stru ctur es. Pr o of. W ell-deﬁnedness of g π k is straightforw ard to chec k. g θ k ( eb ⊗  x y  ) =  eb ⊗ x eb ⊗ y + ∇ T ( eb ) x  =  e ⊗ x e ⊗ y + ∇ T ( e ) θ T k ( b ) x + e ⊗ [ T , θ T k ( b )] x  = g π k ( e ⊗ π T k ( b )  x y  ) . . Its inv e rse is the map e ⊗  x y  7→ e ⊗  x y  − ∇ T ( e )  0 x  , as is chec ked by computation. F o r g θ k , w ell deﬁnedness is mor e o f a sur prise. First observ e that g χ k is the ﬁrst comp onent of g θ k , so w e know this is a well deﬁned top olog ical isomo r phism. In case  − T y y  ∈ Dom T k +1 , we can write v [ k ] p t k v ∗ [ k ]  − g θ k − 1 ( e ⊗ T y ) g θ k − 1 ( e ⊗ y + ∇ T ( e ) T y )  =  − t (1 + t 2 ) − 1 e ⊗ (1 + T 2 ) y (1 + t 2 ) − 1 e ⊗ (1 + T 2 ) y  . F o r such elements, we also have the expres sion θ T k ( b )  − T y y  =  − T (1 + T 2 ) − 1 θ k − 1 ( b )(1 + T 2 ) y (1 + T 2 ) − 1 θ k − 1 ( b )(1 + T 2 ) y  , from whic h well deﬁnedness follows direc tly . Thus, g θ k is well deﬁned and com- pletely b ounded on a dense subset of E k ˜ ⊗ θ k ( G ( T k ⊕ v [ k ] G ( T k ), a nd hence extends to a w e ll deﬁned map on the en tire tensor pro duct module. The f act that g θ k is a top ological isomorphism is proved in a similar fas hio n as for g χ k , by ﬁrst stabilizing and co ns idering the Grassmannian connection. The s ame co mputation as in the pro of o f 5.4.1 then shows that for this connectio n the map is unitary .  46 BRAM M ESLAND Corollary 5. 5.2. L et B b e a C k -algebr a, with deﬁning C k sp e ctr al triple ( H , D ) , p the pr oje ction onto B H and E ⇌ B a C k -mo dule with c onn e ction. Ther e ar e c ompletely b ounde d isomorphisms K B i ( E i ) ∼ = Sob i (1 ⊗ ∇ D ) ∩ K B ( E ) ⊗ p, of involut ive op er ator algebr as, for al l i ≤ k , and similarly for E nd ∗ B i ( E i ) . In p articular K B ( E ) is c ompletely b ounde d ly i somorphic to a C k -algebr a and K B i ( E i ) = K B ( E ) ∩ End ∗ B i ( E i ) . Pr o of. The a lg ebra B k is c o mpletely isometrica lly is omorphic to a closed subalgebra of L k i =0 M 2 i ( B ( H )), via the deﬁning r epresentation π [ k ] . Then by theor e m B.6, p ∈ Sob k ( D ), and by theorem B.8, K B k ( E k ) is completely b oundedly isomor phic to K B k ( E k ) ⊗ π [ k ] ( p ) ⊂ k M i =0 B ( E i ˜ ⊗ B i ( 2 i M j =1 H )) . Cho ose a connection ∇ : E k → E k ˜ ⊗ B k Ω 1 ( B k ). W e hav e K B k ( E k ) ⊗ π [ k ] ( p ) ⊂ Sob k (1 ⊗ ∇ D ), b y th eorem 5.4.1 , and we ha ve to s how that it is closed. Consider the map g π 1 from theor em 5.5 .1. It intert wines the representations a ⊗ 1 and π t 1 , and since K B 1 ( E 1 ) ⊗ π [1] ( p ) is a close d subalge bra o n the left side, it is so on the right side. The same a rgument works for End ∗ B 1 ( E 1 ). Next, ass ume that we ha ve pr oven the result for End ∗ B i − 1 ( E i − 1 ) and K B i − 1 ( E i − 1 ). The o pe r ator space s tructure on B i is given by the represe ntation π [ i ] . Applying theorem 5.5.1 to π i , a nd using the induction h y po thesis on L i − 1 j =0 π j , we see that K B i ( E i ) ⊗ p is a closed subset o f Sob i (1 ⊗ ∇ T ). The same reaso ning applies to End ∗ B i ( E i ). Since K B i ( E i ) is dense in K B ( E ), it is a C k -algebra .  A C k -mo dule with co nnection ( E , ∇ ) can thus b e viewed as a C k - Morita equiv- alence b etw een K B k ( E k ), a nd the ideal h E k , E k i ⊂ B k . Corollary 5. 5.3. L et E ⇌ B b e a C k -mo dule over a C k -algebr a B . The n E nd ∗ B k ( E k ) is sp e ctr al invariant in End ∗ B ( E ) and K B ( E ) is holomorphic al ly dense in K B k ( E k ) . Pr o of. The argument fro m theor em 4.2.4, do e s not dir e ctly apply since E nd ∗ B k ( E k ) need not b e dense in End ∗ B ( E ). When we replace End ∗ B ( E ) by the unital C ∗ - subalgebra A = End ∗ B k ( E k ) , we ﬁnd that End ∗ B k ( E k ) is spectr al in v a riant in A , which is sp ectral inv ariant in End ∗ B ( E ).  6. Correspondences W e hav e seen ho w to emplo y co nnec tio ns as a tool in cons tructing pr o ducts of un bo unded selfadjoint opera tors. This obser v ation leads to the co ns truction of a category of spe ctral triples. The y giv e a no tion of morphism of noncommutativ e geometries, in such a wa y that the b ounded tra nsform induces a functor from corres p o ndences to K K -gr oups. By conside r ing sev eral lev els of diﬀerentiabilit y and smo o thness o n cor resp ondences, one gets s ub ca tegories of corresp ondence s of C k - and smo oth C ∗ -algebra s. BIV ARIANT K -THEOR Y AND CORRESPONDENCES 47 6.1. Al most an ticommuting op erators. In this sec tio n we describ e co nditions on tw o almo st selfadjoint regula r o p e r ators, implying that they induce almost self- adjoint r egular oper ators in each others g raphs. In the case o f selfadjoin t op era tors, this can be used to show that their sum is selfadjo int on the intersection o f their do- mains. In the next section, where we introduce connections, pairs of such op era tors will b e constructed in a natura l wa y . Deﬁnition 6.1 .1. Let E ⇌ B b e a C k -mo dule and s and t almos t selfadjoint regular op era tors in E k . The op er ators s and t almost antic ommute if (1) Ther e exis ts λ > 0 s uch that the ope rators ( s ± λi ) − 1 ( t ± λi ) − 1 and ( s ± λi ) − 1 ( t ∓ λi ) − 1 and their adjoints all hav e the same range. (2) The op erator st + ts , deﬁned on Im ( s ± λi ) − 1 ( t ∓ λi ) − 1 , extends to a n op erator in End ∗ B k ( E k ); W e need the following lemma. Lemma 6. 1.2. L et s : Dom s → E k b e a close d densely deﬁne d op er ator, such that: (1) Ther e ex ist s λ ∈ R \ { 0 } such that s ± λi is surje ct ive and ( s ± λi ) − 1 ∈ End ∗ B k ( E k ) ; (2) Do m s ⊂ Dom s ∗ and s − s ∗ extends to a n op er ator in End ∗ B k ( E k ) . Then s is almost selfadjo int and r e gular in E k . Pr o of. W rite R for the extension of s − s ∗ to all of E k . The op erator s + 1 2 R is symmetric o n Dom s , and x = ( s + 1 2 R ± λi )( s ± λi ) − 1 = 1 + 1 2 R ( s + λi ) − 1 , so, increa sing λ if neces s ary , 0 / ∈ Sp( x ), so b y corollary 5.5.3, x is an invertible op erator in End ∗ B k ( E k ). Therefore s + 1 2 R ± λi is sur jective. Thus s + 1 2 R is selfadjoint r e g ular on Dom s by theor e m 4.5.7.  F o r a pair ( s, t ) of almost anticomm uting ope rators , we can deﬁne Dom χ t 1 ( s ) := { ( e, te ) ∈ G ( t ) : e ∈ Im ( s ± λi ) − 1 ( t ± λi ) − 1 ⊂ Do m s ∩ Domt } , and χ t 1 ( s )( e, te ) := ( se, tse ), for e ∈ Dom χ t 1 ( s ). The nota tion χ t 1 ( s ) indicates the analogy with the bounded case. χ s 1 ( t ) : Do m χ s 1 ( t ) → G ( s ) is deﬁned similarly , b y switching s and t . Prop ositi o n 6.1.3 . L et s and t b e almost ant ic ommuting op er ators. Then χ t 1 ( s ) and χ s 1 ( t ) ar e a lmost self adjoi nt and r e gu lar. Mor e over, the map G ( χ t 1 ( s )) → G ( χ s 1 ( t )) ( e, te, se, ste ) 7→ ( e, s e, te, tse ) , is a t op olo gic al isomorphism of C k -mo dules. Pr o of. First we pro ve χ t 1 ( s ) is almost selfadjoint. By deﬁnition, χ t 1 ( s ) + λi : Dom χ t 1 ( s ) → G ( t ) is surjective. Moreov er since [ t, ( s + λi ) − 1 ] = ( s + λi ) − 1 [ s, t ]( s − λi ) − 1 ∈ E nd ∗ B k ( E k ) , we can write ( χ t 1 ( s ) + λi ) − 1 = χ t 1 (( s + λi ) − 1 ) ∈ E nd ∗ B k ( G ( t )) . 48 BRAM M ESLAND Thu s, b y lemma 6 .1.2, χ t 1 ( s ) is almost selfadjoint and regular in G ( t ). The sta tement on the topolog ical type follo ws by observing that the map G ( χ t 1 ( s )) → G ( χ s 1 ( t )) deﬁned a b ov e ca n b e written as p χ s 1 ( t )     1 0 0 0 0 0 1 0 0 1 0 0 [ s, t ] 0 0 − 1     p χ t 1 ( s ) ∈ Ho m ∗ B k ( G ( χ t 1 ( s )) , G ( χ s 1 ( t ))) , and its inv er se is o btained b y interc hanging p χ s 1 ( t ) and p χ t 1 ( s ) .  Corollary 6. 1.4. F or almost antic ommtuing op er ators s and t and λ, µ suﬃciently lar ge Im ( s ± λi ) − 1 ( t ± µi ) − 1 = Im ( t ± λi ) − 1 ( s ± µ i ) − 1 , Im ( s ∓ λi ) − 1 ( t ± µi ) − 1 = Im ( t ± λi ) − 1 ( s ∓ µ i ) − 1 . Pr o of. It is immediate that Im ( s + λi ) − 1 ( t + λi ) − 1 = Im ( s + λi ) − 1 ( t + µi ) − 1 . The equality Im ( t + λi ) − 1 ( s + λi ) − 1 = Im ( t + µi ) − 1 ( s + λi ) − 1 follows from almost selfadjointness of χ t 1 ( s ).  One may deﬁne almost selfadjoint op er ators χ t j ( s ) in G ( t j ) inductiv ely whenever χ t j − 1 ( s ) a nd t almost an ticommut e in G ( t j − 1 ). W e now turn to the sub ject of the s um o f almost a nticomm uting selfadjoint op erator s in C ∗ -mo dules.T o this end we will use the following p ositivity re sult: Whenever h, k ∈ E nd ∗ B ( E ) a re p o sitive, h has dens e range, a nd h ≤ k , then k has dense range. The reader can co nsult [23] for a pro o f of this statement, which plays a cr ucial r ˆ o le in the subsequent discussion. Lemma 6.1.5. L et ( s, t ) b e a p air of almost antic ommu ting selfadjoi nt op er ators in a C ∗ -mo dule E . F or λ , µ p ositive and suﬃciently lar ge, the op er ators x = ( t − µi ) − 1 − ( s + λi ) − 1 and y = ( t + µi ) − 1 − ( s − λi ) − 1 have dense ra nge. Pr o of. W e show tha t xx ∗ has dense ra nge, which implies x has dense range. W rite a = ( s + λi ) − 1 and b = ( t − µi ) − 1 and compute xx ∗ = ( a − b )( a ∗ − b ∗ ) = aa ∗ + bb ∗ − ab ∗ − ba ∗ . W e kno w that bo th aa ∗ and bb ∗ , a nd hence also aa ∗ + bb ∗ hav e dense range. Now observe that − ab ∗ − ba ∗ = ab ∗ ( λµ − [ s, t ]) ba ∗ ≥ 0 , and therefor e xx ∗ ≥ aa ∗ + bb ∗ , s o xx ∗ has dense range.  Lemma 6.1.6. L et ( s, t ) b e a p air of almost ant ic ommuting op er ators in a C ∗ - mo dule E . Then the sum s + t is close d and symmetric on Dom s ∩ Do m t , and Im ( s + λi ) − 1 ( t + µi ) − 1 is a c or e for s + t . BIV ARIANT K -THEOR Y AND CORRESPONDENCES 49 Pr o of. The sum s + t is symmetric, and it is clos ed on Dom s ∩ Dom t , which can b e seen a s follows. Let x n be a s equence in Im ( t ± λi ) − 1 ( s ± λi ) − 1 ⊂ Dom s ∩ Dom t conv er ging to x ∈ E , and s uch tha t ( s + t ) x n is Cauch y in E k . Then, for y = x n − x m , h ( s + t ) y , ( s + t ) y i = h sy , sy i + h ty , ty i + h sy , ty i + h ty , sy i = h sy , sy i + h ty , ty i + h [ s, t ] y , y i , and since [ s, t ] is b ounded on Im ( t ± λi ) − 1 ( s ± λi ) − 1 , we hav e h [ s, t ]( x n − x m ) , ( x n − x m ) i → 0 for n ≥ m → ∞ . Since the o ther tw o terms ar e p ositive, they must converge to zero as well (since the left hand side do es so). Thus, b oth sx n and tx n are conv e r gent, and since both s and t a re closed, w e ha ve x ∈ Dom s ∩ Dom t and ( s + t ) x n m ust conv er ge to ( s + t ) x . So s + t is closed on Dom s ∩ Dom t , and Im ( t ± λi ) − 1 ( s ± λi ) − 1 is a core for s + t .  Lemma 6.1. 7 . L et D b e a close d symmetric op er ator op er ator in C ∗ -mo dule E . Supp ose ther e exist R ± ∈ End ∗ B ( E ) and λ > max { ( k R ± k + 1) 2 } such that t he op er ators D + R ± ± λi : Do m D → E , have dense r ange in E . The n D ± λi ar e surje ctive and D is selfadjoint and r e gular in E . Pr o of. W rite ˜ D for the clos ed op erator D + R + and b = R + − R ∗ + . F or e ∈ Do m D we can estimate h ( ˜ D + λi ) e, ( ˜ D + λi ) e i = h ˜ D e, ˜ D e i + λ 2 h e, e i + λ h ibe, e i ≥ λ ( λ − k b k ) h e, e i . This shows that ( ˜ D + λi ) − 1 is injectiv e and extends to an op erato r r ∈ End ∗ B ( E ), with (6.26) k r k ≤ 1 p λ ( λ − k b k ) < 1 √ λ < 1 k R + k , bec ause λ > max { ( k R ± k + 1) 2 } ≥ k R + k 2 + k b k + 1. Let e ∈ E b e arbitrar y and e n ∈ Im ( ˜ D + λi ) b e a sequence conv erg ing to e . Then ( ˜ D + λi ) − 1 e n → re and ˜ D ( ˜ D + λi ) − 1 e = (1 − λi ( ˜ D + λi ) − 1 ) e → e − λir e. Since D is closed we have r e ∈ Dom D and ( ˜ D + λi ) r e = e . That is, r = ( ˜ D + λi ) − 1 and ˜ D + λi : Dom D → E is bijectiv e. Now k R + ( D + R + + λi ) − 1 ) k < 1 b y (6.26), so we see that ( D + λi )( D + R + + λi ) − 1 = 1 − R + ( D + R + + λi ) − 1 , is inv ertible. Hence D + λi : Dom D → E is bijective. Using D + R − − λi , one shows in the same wa y that D − λi is bijective to o, so D is selfadjoint a nd regular .  Theorem 6. 1.8. L et ( s, t ) b e a p air of almost antic ommuting op er ators in a C ∗ - mo dule E . Then t he sum s + t is selfadjo int and r e gular on Dom s ∩ Do m t . Pr o of. The ope rator s + t is closed and symmetric by lemma 6.1.6. Consider the op erator s x and y fro m lemma 6.1.5. W e can factor thes e oper ators a s x = ( s + t + ( µ − λ ) i − ( s + λi ) − 1 ([ s, t ] − 2 λµ ))( s − λi ) − 1 ( t − µi ) − 1 , y = ( s + t + ( λ − µ ) i − ( s − λi ) − 1 ([ s, t ] − 2 λµ ))( s − µi ) − 1 ( t − λi ) − 1 , 50 BRAM M ESLAND by a standar d algebraic computation. By le mma 6.1.5, x a nd y hav e dense ra nge, and therefor e the op erato rs s + t ± ( µ − λ ) i − ( s ± λi ) − 1 ([ s, t ] − 2 λµ ) : Dom s ∩ Dom t → E hav e dense range. Cho osing λ, µ pos itive, and such that λ > (2 + 2 µ ) 2 > k [ s, t ] k , we ﬁnd that R ± = − ( s ± λi ) − 1 ([ s, t ] − 2 λµ ) ∈ E nd ∗ B ( E ) , hav e nor m k R ± k < k [ s, t ] k λ + 2 µ ≤ 1 + 2 µ, and λ − µ > (2 + 2 µ ) 2 ≥ ( k R ± k + 1) 2 . Th us we can apply lemma 6.1 .7 to the closed symmetric oper a tor s + t , the op erator s R ± and the co nstant λ − µ , t o ﬁnd that s + t is selfadjoint and regular.  Note that for a lmost s elfadjoint almost anticomm uting op er ators, a lmost se lfa d- joint ness o f the sum is not guaranteed by the ab ov e considera tions. Next, we consider triples o f a lmost a nt icommuting op e rators in a C ∗ -mo dule. A tr iple of ope r ators ( s, t, ∂ ) is said to almost antic ommute if eac h pair of them almost a nt icommutes and if Im ( s + λi ) − 1 ( t + λi ) − 1 ( ∂ + λi ) − 1 = Im ( ∂ + λi ) − 1 ( s + λi ) − 1 ( t + λi ) − 1 . Note tha t this implies tha t any order of resolven t pro ducts will hav e the same rang e, using that pairs almost anticomm ute. Prop ositi o n 6.1.9. L et ( s, t, ∂ ) i s a n almo st antic ommut ing triple of almost self- adjoint re gular op er ators in a C ∗ -mo dule E . Su pp ose t hat s + t is almost selfadjoint r e gular with c or e Im ( s + λi ) − 1 ( t + λi ) − 1 .Then s + t and ∂ almost antic ommu t e. Pr o of. F or notational co nv enenience w e write a = ( ∂ + λi ) − 1 , b = ( s + λi ) − 1 , c = ( t + λi ) − 1 , d = ( s + t + λi ) − 1 . W e hav e to show that (6.27) Im ( s + t + λi ) − 1 ( ∂ + λi ) − 1 = Im ( ∂ + λi ) − 1 ( s + t + λi ) − 1 , and that the commut ator [ s + t, ∂ ] , is bounded on this se t. No te that Im ( ∂ + λi ) − 1 ( s + t + λi ) − 1 = Im ( ∂ + λi ) − 1 ( s + λi ) − 1 ∩ Im ( ∂ + λi ) − 1 ( t + λi ) − 1 = Im ( s + λi ) − 1 ( ∂ + λi ) − 1 ∩ Im ( t + λi ) − 1 ( ∂ + λi ) − 1 ⊂ Im ( s + t + λi ) − 1 , and that [ ∂ , s + t ] is b o unded on this subset. Consider s + t as an o p erator in the graph G ( ∂ ), deﬁned o n the abov e domain, and denote it b y χ ∂ 1 ( s + t ). Then χ ∂ 1 ( s + t ) is a closed o p erator: Suppo se (1) adx n ∂ − → ax, (2) ( s + t ) adx n ∂ − → ay , BIV ARIANT K -THEOR Y AND CORRESPONDENCES 51 where ∂ − → means conv er gence in G ( ∂ ). (1) implies that dx n is a convergen t sequence in E . Moreov er we hav e ( s + t ) adx n = − a ∗ ( s + t ) dx n + [ s, a ] dx n + [ t, a ] dx n , from which it follows that − a ∗ ( s + t ) dx n is convergen t (in E ). Mor eov e r , since [ s, a ] = sa + a ∗ s o n Dom s, [ t, a ] = ta + a ∗ t o n Do m t, in particula r on Im d = Dom s ∩ Do m t we get [ s, a ] = a ∗ [ s, ∂ ] a, [ t, a ] = a ∗ [ t, ∂ ] a. Therefore the term [ s, a ] dx n + [ t, a ] dx n , is convergen t in G ( ∂ ), and hence a ∗ ( s + t ) dx n is so to o. Then from a ∗ ( s + t ) dx n = a ∗ x n − ia ∗ dx n , it follows that a ∗ x n is co nvergen t in G ( ∂ ), whic h means that x n is a con vergen t sequence in E . F rom this it follows rea dily that χ ∂ 1 ( s + t ) is clo sed in G ( ∂ ). It is almost symmetric on its do main since χ ∂ 1 ( s ) and χ ∂ 1 ( t ) are almos t s elfadjoint. Thus it remains to show that χ ∂ 1 ( s + t ) + λi ha s dense range for some λ . T o this end we use that Im abc ⊂ Do m χ ∂ 1 ( s + t ) . Then since ( s + t + λi ) abc = a ∗ ( − s − t + λi ) bc + [ s, a ] b c + [ t, a ] bc = a ∗ ( − s − t + λi + [ s, ∂ ] a + [ t, ∂ ] a ) bc, and [ s, ∂ ] a + [ t, ∂ ] a is bo unded, fo r λ large e nough we hav e that Im ( − s − t + λi + [ s, ∂ ] a + [ t, ∂ ] a ) bc, is dense in E . Hence Im a ∗ ( − s − t + λi + [ s, ∂ ] a + [ t, ∂ ] a ) bc, is dense in G ( ∂ ). Th us χ ∂ 1 ( s + t ) is almost s e lfa djoint G ( ∂ ). This implies (6.27), and the c o mmut ator pro p e r ties a re immediate, so s + t a nd ∂ almost anticomm ute.  By the sa me metho ds, w e c an prove the following pro po sition. Prop ositi o n 6.1.10. L et s, t b e almost antic ommut ing almost selfadjoint op er ators in a C k -mo dule E k . Supp ose s + t is almost selfadjoi nt on Do m s ∩ Dom t , with c or e Im ( s + λi ) − 1 ( t + λi ) − 1 . Then (1) χ s ( s + t ) is almost selfadj oint in G ( s ) and χ s ( s + t ) = χ s ( s ) + χ s ( t ) , i.e. Dom χ s ( s + t ) = Dom χ s ( s ) ∩ Dom χ s ( t ) . (2) χ t ( s + t ) is almost selfa djoint i n G ( t ) and χ t ( s + t ) = χ t ( s ) + χ t ( t ) , i.e. Dom χ t ( s + t ) = Dom χ t ( s ) ∩ Dom χ t ( t ) . (3) Do m ( s + t ) 2 = Do m s 2 ∩ Im ( s + λi ) − 1 ( t + λi ) − 1 ∩ Do m t 2 . 52 BRAM M ESLAND Pr o of. Statements (1) and (2) are prov ed as in the previous prop osition. F or (3), note that Dom ( s + t ) 2 = ( s + t + λi ) − 1 Im ( s + λi ) − 1 ∩ Im ( t + λi ) − 1 = Im ( s + λi ) − 1 ( s + t + λi ) − 1 ∩ Im ( t + λi ) − 1 ( s + t + λi ) − 1 = Im ( λ 2 + s 2 ) − 1 ∩ Im ( s + λi ) − 1 ( t + λi ) − 1 ∩ Im ( λ 2 + t 2 ) − 1 , where the second equality follows by using (1) and (2).  Suppo se we hav e a pair ( s, t ) of a lmost selfadjoint op era to rs in E k , who se sum s + t is almo st selfadjoint on Dom s ∩ Dom t . The graphs of s and t b oth map completely bo undedly to E k , b y pro jection onto the ﬁrs t factor. Hence the pullback G ( s ) ∗ G ( t ) is deﬁned, as the universal solution to the diag ram G ( s ) ∗ G ( t ) ✲ G ( s ) G ( t ) ❄ ✲ E k . ❄ It can be ident iﬁed (as a top olo gical C k -mo dule) with the submodule of G ( s ) ⊕ G ( t ) given by G ( s ) ∗ G ( t ) := { ( e, se, e, te ) : e ∈ Dom s ∩ Dom t } . Prop ositi o n 6.1.11 . If s and t ar e almost antic ommuting almost selfadjoi nt r e gu- lar op er ators in E k such t hat s + t is almost selfadjoint r e gular on Dom s ∩ Do m t ⊂ E k , with c or e Im ( s + λi ) − 1 ( t + λi ) − 1 then ther e is a t op olo gic al isomorphism of C k -mo dules g : G (( s + t )) ∼ − → G ( s ) ∗ G ( t ) ( e, ( s + t ) e ) 7→ ( e, se, e , te ) . Pr o of. By (3) of prop ositio n 6.1.1 0, Dom ( s + t ) 2 = Dom s 2 ∩ Im ( s + λi ) − 1 ( t + λi ) − 1 ∩ Dom t 2 , and since s, t almost ant icommute, [ s, t ] is bounded o n Dom ( s + t ) 2 and so s 2 + t 2 is a bounded p erturbation of ( s + t ) 2 . Hence it is almost selfadjoint reg ular on Dom ( s + t ) 2 , and the op e rator λ 2 + s 2 + t 2 is a bijection for λ suﬃcien tly large. In the following w e take λ = 1, which can alwa ys be ac hieved by rescaling. The mo dule G ( s ) ∗ G ( t ) ⊂ L 4 j =1 E k is the range o f the (non-selfa djoint ) idemp otent q :=     a as a at sa s as sa sat a as a at ta tas ta tat     , where a = (1 + s 2 + t 2 ) − 1 . Thus, b y cor ollaries 5.5 .2 and 4.2.5 there is a pro jectio n p with p 4 M j =1 E k = G ( s ) ∗ G ( t ) , BIV ARIANT K -THEOR Y AND CORRESPONDENCES 53 and the map g ca n b e written as g = p     b ( s + t ) b sb s ( s + t ) b b ( s + t ) b tb t ( s + t ) b     p s + t , where b = (1 + ( s + t ) 2 ) − 1 and p s + t the W oro nowicz pro jection. Moreov e r , w e hav e g − 1 = p s + t  1 2 0 1 2 0 0 1 0 1  p, showing that g is a topolo gical iso morphism.  6.2. The pro duct of transv erse mo dule s. W e now show that the op erator s S ⊗ 1 and 1 ⊗ ∇ T almost anticomm ute in the Sobolev mo dules of 1 ⊗ ∇ T . F rom now on, write s = S ˜ ⊗ 1 and t = 1 ˜ ⊗ ∇ T . The resolvents of s a nd t satisfy the following crucia l compa tibilit y . Lemma 6.2.1. L et ( E , S ) and ( F , T ) b e tr ansverse C k ( A, B ) a nd ( B , C ) bimo d- ules r esp e ctively, and ∇ : E k → E k ˜ ⊗ B k Ω 1 ( B k ) a tr ansverse C n -c onne ction. The C k -endomorphisms ( t ∓ λi ) − 1 ( s ± λi ) − 1 , ( t ± λi ) − 1 ( s ± λi ) − 1 , ( s ∓ λi ) − 1 ( t ± λi ) − 1 , ( s ± λi ) − 1 ( t ± λi ) − 1 , al l h ave the same r ange in G ( S i ) ˜ ⊗ B j G ( T j ) , for i ≤ n − 1 , j ≤ k − 1 . Pr o of. Denote by pr S : G ( S i +1 ) → G ( S i ) and pr T : G ( T j +1 ) → G ( T j ) the ad- joint able o p er ators given by pro jection on the ﬁr st co or dinate of the gra ph. pr T is a B j -mo dule map, and hence by theo rem 3 .3.10, pr S ⊗ pr T : G ( S i +1 ) ˜ ⊗ B j +1 G ( T j +1 ) → G ( S i ) ˜ ⊗ B j G ( T j ) , is an a djointable op erator. W e will show tha t a ll op er ators hav e range Im pr S ⊗ pr T . F o r a ny λ > 0 , ( s ± λi ) − 1 maps G ( S i ) ˜ ⊗ B j G ( T j ) bijectively onto Dom S i +1 ⊗ 1, which is in bijection wit h G ( S i +1 ) ˜ ⊗ B j G ( T j ) ∼ = G (1 ⊗ ∇ i +1 T ) j . Since (1 ⊗ ∇ i +1 T ) j is almost selfa djoint in G (1 ⊗ ∇ i +1 T ) j − 1 , ( t ± λi ) − 1 maps this mo dule bijectiv ely onto Dom t ⊂ G (1 ⊗ ∇ i +1 T ) j − 1 for λ suﬃciently la rge. This domain in turn is in bijection with G ( S i +1 ) ˜ ⊗ B j G ( T j ), by theor em 5.4.1. The diagram Dom s ( t ± λi ) − 1 ✲ G ( S i ) ˜ ⊗ B j G ( T j ) G ( s ) ❄ ✲ G ( S i +1 ) ˜ ⊗ B j +1 G ( T j +1 ) , pr S ⊗ pr T ✻ commutes, which means we have shown Im ( t ± λi ) − 1 ( s ± λi ) − 1 = Im pr S ⊗ pr T . 54 BRAM M ESLAND The map r : G ( S i ) → G ( S i +1 ) e 7→ (( S + λi ) − 1 e, S ( S + λi ) − 1 e ) is a top ologica l is omorphism, and hence, by theo rem 3 .3 .10, r ⊗ 1 : G ( S i ) ˜ ⊗ B j G ( T j ) → G ( S i +1 ) ˜ ⊗ B j G ( T j ) , is a top ologica l is omorphism. Moreover the diagram Dom t ( s + λi ) − 1 ✲ G ( S i ) ˜ ⊗ B j G ( T j ) G ( S i ) ˜ ⊗ B j +1 G ( T j +1 ) ❄ r ⊗ 1 ✲ G ( S i +1 ) ˜ ⊗ B j +1 G ( T j +1 ) , pr S ⊗ pr T ✻ commutes, where the down ward arr ow is the bijection Dom t → G ( t ) i comp osed with the map from theorem 5.4.1. This prov es that Im pr S ⊗ pr T = Im ( s ± λi ) − 1 ( t ± λi ) − 1 .  Given a selfadjoint regula r C k -op erator S in E k , we get naturally induced o p- erators S ⊗ 1 in a ll the mo dules E i ˜ ⊗ B i G ( T i ), for i ≤ k . Although these op erator s need not b e almost selfa djoint they are still r egular. Lemma 6.2. 2 . L et E , F b e C k -mo dules over B , g : E k → F k a t op olo gic al iso- morphism, and D an almost selfadjo int r e gular op er ator in E k . Then g D g − 1 is r e gular in F k . Pr o of. Since D is a lmo st selfadjoint, D ± λi : D → E k are bijections, and ( D ± λi ) − 1 ∈ End ∗ B k ( E k ). Ther efore both g ( D + λi ) − 1 g − 1 and g − 1 ∗ ( D ∗ − λi ) − 1 g ∗ hav e dense ra nge in F k and by coro llary 4.5.6, their inv erses ar e reg ular. Th us g D g − 1 = g ( D + λi ) g − 1 − λi, is a bo unded pertur bation o f a regular o pe rator, hence regular .  Prop ositi o n 6.2.3. L et ( E , S ) and ( F , T ) b e tr ansverse C k ( A, B ) and ( B , C ) bi- mo dules r esp e ctively, and ∇ : E k → E k ˜ ⊗ B k Ω 1 ( B k ) a t ra nsverse C k -c onne ction. F or i, j ≤ k − 1 , write t = 1 ⊗ ∇ i T . The op er ators χ t j ( s ) and t j almost antic ommut e in G ( t j ) . Conse qu en tly, the op er ators S i +1 ⊗ 1 ar e r e gular in e ach G ( S i ) k ˜ ⊗ B k G ( T j ) k , with gr aph G ( S i +1 ) k ˜ ⊗ B k G ( T j ) k . Pr o of. By theorem 5.4 .1, the mo dules G ( S i ) j ˜ ⊗ B j G ( T j ) ar e top olo g ically isomor phic to the Sobo lev mo dules G ((1 ⊗ ∇ i T ) j ) o f 1 ⊗ ∇ i T , an almos t selfadjoint op erato r in G ( S i ) k ˜ ⊗ B k F k ∼ = G ( s i ). By lemma 6.2.1 Im ( s ± λi ) − 1 ( t ± λi ) − 1 = Im ( t ± λi ) − 1 ( s ± λi ) − 1 , in G ( S i ) ˜ ⊗ B k F k , and [ s, t ] = [ ∇ i , S ] ⊗ 1, so s and t almost anticomm ute cf. de ﬁ- nition 6.1.1, so χ t 1 s is almost s elfadjoint in G (1 ⊗ ∇ i T ) b y 6.1.3. The topo logical isomorphism g χ : G ( S i ) ˜ ⊗ B 1 G ( T ) → G (1 ⊗ ∇ i T ) , BIV ARIANT K -THEOR Y AND CORRESPONDENCES 55 satisﬁes g χ χ t 1 ( s )( g χ ) − 1 = S ⊗ 1. So by lemma 6.2 .2, S ⊗ 1 is reg ular in G ( S i ) ˜ ⊗ B 1 G ( T ), and its graph is G ( S i +1 ) ˜ ⊗ B 1 G ( T ). P ro ceeding by induction, supp os e we hav e shown that χ t j ( s ) is almost selfadjoint in G ((1 ⊗ ∇ i T ) j ) and hence S ⊗ 1 is regular in G ( S i ) ⊗ B j G ( T j ). By lemma 6.2.1 we hav e Im ( s ± λi ) − 1 ( t ± λi ) − 1 = Im ( t ± λi ) − 1 ( s ± λi ) − 1 , in G ( S i ) ⊗ B j G ( T j ) a nd hence also in G ((1 ⊗ ∇ i T ) j ), a s g χ j int ertwines thes e op er- ators. Moreov er [ χ t j ( s ) , t ] = g χ j ([ ∇ i , S ] ⊗ 1)( g χ j ) − 1 , which is b ounded o n Im ( s ± λi ) − 1 ( t ± λi ) − 1 , so χ t j ( s ) a nd t almo st anticomm ute in G ((1 ⊗ ∇ i T ) j ), a nd χ t j +1 ( s ) is almos t selfadjoint in G ((1 ⊗ ∇ i T ) j +1 ), by 6.1 .3. The top ological isomorphism g χ j +1 int ertwines χ t j +1 and S ⊗ 1, so the latter oper ator is regular in G ( S i ) ˜ ⊗ B j +1 G ( T j +1 ) by lemma 6.2.2.  Lemma 6.2.4. L et ( E , S, ∇ ) and ( F , T , ∇ ′ ) b e C k bimo dules with tr ansverse C k - c onne ction. We have [ S ⊗ 1 , 1 ⊗ ∇ i ∇ ′ j ] = [ S, ∇ i ] ⊗ 1 , [1 ⊗ ∇ i T , 1 ⊗ ∇ i ∇ ′ j ] = 1 ⊗ ∇ i [ ∇ j , T ] + [ ∇ i ∇ ′ j , ∇ iT ] , wher e the le ft hand sid es ar e deﬁne d on Dom S ⊗ 1 ⊂ G ( S i ) k ˜ ⊗ B k G ( T j ) k and Dom 1 ⊗ ∇ i T ⊂ G ( S i ) k ˜ ⊗ B k G ( T j ) k , r esp e ctively. Thus, t hese c ommutators ex tend to c ompletely b ounde d maps G ( S i ) k ˜ ⊗ B k G ( T j ) k → G ( S i ) k ˜ ⊗ B k G ( T j ) k ˜ ⊗ C k Ω 1 ( C k ) . Pr o of. The conditions imply we hav e transverse C k +1 − i connections ∇ i : G ( S i ) k → G ( S i ) k ˜ ⊗ B k Ω 1 ( B k ) , ∇ ′ i : G ( T j ) k → G ( T j ) k ˜ ⊗ C k Ω 1 ( C k ) , with the prop erty that [ ∇ i , S ], [ ∇ ′ j , T ] a re b ounded endomorphisms of the resp ective mo dules. These as well deﬁne pro duct connections 1 ⊗ ∇ i ∇ ′ j on G ( S i ) k ˜ ⊗ B k G ( T j ) k , i, j ≤ k . W e show such connections boundedly comm ute wit h S ⊗ 1 and 1 ⊗ ∇ T . Since [1 ˜ ⊗ ∇ ∇ ′ , S ˜ ⊗ 1 + 1 ˜ ⊗ ∇ T ] = [1 ˜ ⊗ ∇ ∇ ′ , S ˜ ⊗ 1] + [1 ˜ ⊗ ∇ ∇ ′ , 1 ˜ ⊗ ∇ T ] , and [1 ˜ ⊗ ∇ ∇ ′ , S ˜ ⊗ 1] = [ ∇ , S ] ˜ ⊗ 1, which is completely bounded, we compute ( − 1) ∂ e [1 ˜ ⊗ ∇ ∇ ′ , 1 ˜ ⊗ ∇ T ]( e ⊗ f ) to ﬁnd e ⊗ [ ∇ ′ , T ] f + ∇ ∇ ′ ( e ) T f + 1 ˜ ⊗ ∇ ∇ ′ ( ∇ T ( e ) f ) − ∇ T ( e ) ∇ ′ ( f ) − 1 ˜ ⊗ ∇ T ( ∇ ∇ ′ ( e ) f ) . The ﬁrst ter m is completely bo unded, and in working out the las t fo ur terms write ∇ ( e ) = P e i ⊗ db i . Then ∇ ∇ ′ ( e ) T f = X e i ⊗ [ ∇ ′ , b i ] T f , (6.28) ∇ T ( e ) ∇ ′ ( f ) = X e i ⊗ [ T , b i ] ∇ ′ ( f ) , (6.29) 1 ˜ ⊗ ∇ ∇ ′ ( ∇ T ( e ) f ) = X e i ⊗ ∇ ′ [ T , b i ] f + ∇ ∇ ′ ( e i )[ T , b i ] f , (6.30) 1 ˜ ⊗ ∇ T ( ∇ ∇ ′ ( e ) f ) = X i e i ⊗ T [ ∇ ′ , b i ] f + ∇ T ( e i )[ ∇ ′ , b i ] f . (6.31) 56 BRAM M ESLAND Combining 6 .28,6.29 and the ﬁrst terms on the righ t hand sides of 6.30 a nd 6.31 give a term X i e i ⊗ [[ ∇ ′ , T ] , b i ] f = ∇ [ ∇ ′ ,T ] ( e ) f , and the terms remaining fro m 6.30 and 6.31 give a term ( ∇ ∇ ′ ∇ T − ∇ T ∇ ∇ ′ )( e ⊗ f ) . Thu s, we hav e shown that [1 ˜ ⊗ ∇ ∇ ′ , 1 ˜ ⊗ ∇ T ] = 1 ˜ ⊗ ∇ [ ∇ ′ , T ] + [ ∇ ∇ ′ , ∇ T ] , which is a completely b o unded map G ( S i ) k ˜ ⊗ B k G ( T j ) k → G ( S i ) k ˜ ⊗ B k G ( T j ) k ˜ ⊗ C k Ω 1 ( C k ).  Theorem 6.2.5. L et k ≥ 1 , and A, B , C b e C k -algebr as, ( E , S, ∇ ) and ( F , T , ∇ ′ ) tr ansverse C k -bimo dules with c onne ction, fo r ( A, B ) and ( B , C ) r esp e ctively. If ∇ is tr ansverse C 1 , then the op er ator S ⊗ 1 + 1 ⊗ ∇ T is selfadjoint a nd r e gular in E k ˜ ⊗ B k F k . Pr o of. Since ∇ is transverse C 1 the op erator s s = S ⊗ 1 and t = 1 ⊗ ∇ T almost anticomm ute in E ˜ ⊗ B F , and hence by theorem 6.1.8 S ⊗ 1 + 1 ⊗ ∇ T is selfadjoint and regula r on Dom s ∩ Dom t . Since b oth s and t are C 1 , s + t ma ps the C 1 - domain Dom s ∩ Dom t ⊂ E 1 ˜ ⊗ B 1 F 1 int o E 1 ˜ ⊗ B 1 F 1 . T o s how that s + t is selfadjoint in E k ˜ ⊗ B k F k , it suﬃces to show that ( s + t + i ) − 1 is an element o f End ∗ C k ( E k ˜ ⊗ B k F k ). Let k = 1 , and ( C , H , D ) be the deﬁning C 1 -sp ectral triple f or C . Consider the map g π 1 : E 1 ˜ ⊗ B 1 F 1 ˜ ⊗ π ( H ⊕ H ) → E ˜ ⊗ B F ˜ ⊗ C ( H ⊕ H ) , from theor em 5.5 .1. W rite ∂ := 1 ⊗ 1 ⊗ ∇ ∇ ′ D . W e have g End ∗ B 1 ( E 1 ˜ ⊗ B 1 F 1 ) g − 1 ⊂ Sob 1 ( ∂ ) , as a closed subalgebra. Thus it suﬃces to show that under this iso morphism ( s + t + i ) − 1 ∈ So b 1 ( ∂ ) . This means w e hav e to show that ( s + t + i ) − 1 preserves the domain of ∂ , and that [( s + t + i ) − 1 , ∂ ] is b ounded on the domain. By constr uction, the op e r ators ( s, t, ∂ ) form a n almost an ticomm uting triple in the Hilb ert space E ˜ ⊗ B F ˜ ⊗ C ( H . Moreover s + t is s elfadjoint with core Im ( s + λi ) − 1 ( t + λi ) − 1 . Th us , by prop osition 6.1 .9, we ﬁnd that s + t almost anticomm utes with ∂ , a nd in particular that ( s + t + i ) − 1 ∈ Sob 1 ( ∂ ). Pro c eeding by induct ion on k , ( H , D ) is the deﬁning C k sp ectral triple for C , and suppo se we have shown that ( i + s + t ) − 1 ∈ So b k − 1 ( ∂ ), a nd that Im χ ∂ k − 1 (( s + λi ) − 1 ( t + λi ) − 1 ) , is a core for χ ∂ k − 1 ( s + t ). Using the iso morphism g π k : E k ˜ ⊗ B k F k ˜ ⊗ π k ( H k ⊕ H k ) → E k ˜ ⊗ B k F k ˜ ⊗ θ k − 1 ( H k − 1 ⊕ H k − 1 ) , where H k is the k -th Sob o lev space of D , we see by theo rem 5.4.1 that this yields t wo copies of the k − 1-th Sob elev space of ∂ . No w applying prop os ition 6.1.9 again to χ ∂ k − 1 ( s ) , χ ∂ k − 1 ( t ) a nd ∂ k − 1 , we see that ( s + t ± i ) − 1 ∈ So b k ( ∂ ). Hence the op erator S ⊗ 1 + 1 ⊗ ∇ T is se lfa djoint a nd reg ular in E k ˜ ⊗ B k F k .  BIV ARIANT K -THEOR Y AND CORRESPONDENCES 57 Since the pro duct op er a tor is C k , its Sob olev chain can b e canonica lly smo o thened. Also, it is ob viously transverse C 1 . W e now pro ceed to show hig her order tra nsverse smo othness. Lemma 6.2. 6 . L et k ≥ 1 , and A, B , C b e C k -algebr as, ( E , S, ∇ ) and ( F , T , ∇ ′ ) tr ansverse C k -bimo dules with c onne ction, fo r ( A, B ) and ( B , C ) r esp e ctively. If ∇ is tr ansverse C i , with i ≥ 1 , then for al l j ≤ k , the op er ator S i ⊗ 1 + 1 ⊗ ∇ i − 1 T j is r e gular in G ( S i − 1 ) k ˜ ⊗ B k G ( T j − 1 ) k . Mor e over its gr aph is top olo gic al ly isomorphic to G ( S i ) k ˜ ⊗ B k G ( T j − 1 ) k ∗ G ( S i − 1 ) k ˜ ⊗ B k G ( T j ) k . Pr o of. F or i = 1, j = 1, the ope r ator S ⊗ 1 + 1 ⊗ ∇ T is selfadjoint, r egular and C k by theor em 6.2.5. Now prop o s ition 6.1.11 gives the gra ph isomorphism G ( S ⊗ 1 + 1 ⊗ ∇ T ) k → G ( S ) k ⊗ B k F k ∗ E k ˜ ⊗ B k G ( T ) k . Moreov er, s = S ⊗ 1, t = 1 ⊗ ∇ T and s + t are almost selfadjoin t in G ( s ) and G ( t ) b y prop osition 6.1.10. Next we pro ceed b y induct ion on j ≤ k . Supp ose w e hav e shown that S ⊗ 1 , 1 ⊗ ∇ T j − 1 and S ⊗ 1 + 1 ⊗ ∇ T j − 1 are almo s t selfadjoint regular in G (1 ⊗ ∇ T ) k j − 2 , which is top ologic a lly isomorphic to E k ˜ ⊗ B k G ( T j − 2 ) k . Since the connection is transverse C 1 , [ S , 1 ⊗ ∇ T j − 1 ] is b ounded in these mo dules, and prop ositio n 6.1.10 no w gives that S ⊗ 1 , 1 ⊗ ∇ T j and their sum ar e almost selfadjoint regula r in G (1 ⊗ ∇ T ) k j − 1 . Using the propositio n 6.1.11 and theorems 5.4.1 and 6.2.3 gives G ( S ⊗ 1 + 1 ⊗ ∇ T j ) k ∼ − → G ( S ) k ⊗ B k G ( T j − 1 ) k ∗ E k ˜ ⊗ B k G ( T j ) k . W e pro ceed by induction on i , so supp os e we have proven, that for a trans- verse C i -connection, S i ⊗ 1 , 1 ⊗ ∇ i − 1 T j and S i ⊗ 1 + 1 ⊗ ∇ i − 1 T j are almost self- adjoint in the Sobolev mo dule G (1 ⊗ ∇ i − 1 T ) k j − 1 which is to po logically isomor - phic to G ( S i − 1 ) k ˜ ⊗ B k G ( T j − 1 ) k . Since the connection is trans verse C i +1 the com- m utator [ S i +1 ⊗ 1 , 1 ⊗ ∇ i T ] = [ S i +1 , ∇ i ] is b ounded, and by prop o s ition 6.1.10 S i +1 ⊗ 1 , 1 ⊗ ∇ i T j and S i +1 ⊗ 1 + 1 ⊗ ∇ i T j give rise to almost selfa djoint op erator s in the gra ph of S i +1 ⊗ 1, v iewed as an op erato r in G (1 ⊗ ∇ i T ) j . By theo r ems 5.4.1 and 6.2.3, this gr aph is isomorphic to G ( S i +1 ) k ˜ ⊗ B k G ( T j ) k . Combining t his with prop osition 6 .1 .11 gives the graph is omorphism G ( S i +1 ⊗ 1 + 1 ⊗ ∇ i T j ) k ∼ − → G ( S i +1 ) k ˜ ⊗ B k G ( T j − 1 ) k ∗ G ( S i ) k ˜ ⊗ B k G ( T j ) k .  The next result can b e regar ded as a type of K ¨ unneth for m ula for smo o th pro d- ucts. Theorem 6.2.7. L et k ≥ 1 , and A, B , C b e C k -algebr as, ( E , S, ∇ ) and ( F , T , ∇ ′ ) tr ansverse C k -bimo dules for ( A, B ) and ( B , C ) r esp e ctively. F or al l i ≤ k , ther e ar e natur al top olo gic al isomorphi sms g i : G (( S ⊗ 1 + 1 ⊗ ∇ T ) i ) i ∼ − → i ∗ j =0 G ( S j ) i ˜ ⊗ B i G ( T i − j ) i , 58 BRAM M ESLAND wher e the suc c essive pul lb acks ar e over the maps pr S 1 ⊗ pr T 1 . Mor e over g i ◦ χ s + t i ( a ) = ( a ⊗ 1) ◦ g i , for all i ≤ k , and c onse quently A i → Sob i i ( S ⊗ 1 + 1 ⊗ ∇ T ) c ompletely b ounde d ly, and the c onne ction 1 ⊗ ∇ ∇ ′ is tr ansverse C k . That i s ( E ˜ ⊗ B F , S ⊗ 1 + 1 ⊗ ∇ T , 1 ⊗ ∇ ∇ ′ ) is a t r ansverse C k -bimo dule with c onne ction. Pr o of. W rite s = S ⊗ 1 , t = 1 ⊗ ∇ T . F ro m propo sition 6 .1.11 we get that G ( s + t ) ∼ = G ( s ) ∗ G ( t ). This is the theorem for k = 1. Supp ose the theor em has b een pro ven for k . The map g k ⊕ g k : G ( s + t ) k k ⊕ G ( s + t ) k k → 2 M i =1 k ∗ j =0 G ( S j ) k ˜ ⊗ B k G ( T k − j ) k , maps G ( s + t ) k +1 k +1 onto the g r aphs G ( s + t ) ⊂ G ( S j ) k +1 ˜ ⊗ B k +1 G ( T k − j ) k +1 . By lemma 6.2.6, these gra phs are iso morphic to G ( S j +1 ) k +1 ˜ ⊗ B k +1 G ( T k − j ) k +1 ∗ G ( S j ) k +1 ˜ ⊗ B k +1 G ( T k +1 − j ) k +1 . Eliminating the double terms from the pullback, this gives the theo r em for k + 1 . Now let a ∈ A k +1 , and assume b y induction that g k χ s + t k ( a ) = ( a ⊗ 1) g k (this is obvious for k = 1 ). Then g k [ s + t, χ s + t k ( a )] g − 1 k = [ S ⊗ 1 + 1 ⊗ ∇ T , a ⊗ 1 ] = [ S, a ] ⊗ 1 + [ ∇ , a ] ⊗ 1 , in each G ( S j ) ˜ ⊗ B i G ( T k − j ). This is a bounded op erator in G ( S j ) ˜ ⊗ B k +1 G ( T k − j ) k +1 , similarly for a ∗ , so by lemma 4.1.6 a ∈ Sob k +1 k +1 ( s + t ). It is immediate that then g k +1 χ s + t k +1 ( a ) = a ⊗ 1 g k +1 , and the map A k +1 → Sob k +1 k +1 ( s + t ) is c o mpletely bo unded. The statement on the connection 1 ⊗ ∇ ∇ ′ follows b y a simila r arg u- men t applying 6.2.4.  As a consequence, we s ee that for k ≥ 1, tr ansverse C k -triples ( E , S, ∇ ), with C k -connection can b e comp osed according to the r ule ( E , S, ∇ ) ◦ ( F , T , ∇ ′ ) := ( E ˜ ⊗ B F , S ⊗ 1 + 1 ⊗ ∇ T , 1 ⊗ ∇ ∇ ′ ) , and that this comp ositio n is asso ciative up to unitar y equiv alence inducing topo - logical iso morphisms on the g raphs and smo oth structures. 6.3. The K K -pro duct. Now w e establish that compact res o lven ts are prese rved under taking pro ducts. Then we will see that the pro duct op er ator satisﬁes Kucerovsky’s conditions for an unbounded Kas pa rov pro duct. Thus, for smo o th mo dules the K K -pro duct is g iven by an explicit a lgebraic formula. Let us put the pieces to - gether. Lemma 6.3 .1. L et s, t b e selfadjoint re gular op er ators on a C k -mo dule E , and R, a ∈ End ∗ B k ( E k ) with R ∈ End ∗ B k ( E k ) a selfa djoint element. If a ( s + i ) − 1 ( t + i ) − 1 ∈ K B k ( E k ) , then a ( s + i ) − 1 ( t + R + i ) − 1 ∈ K B k ( E k ) . Pr o of. One has the identit y a ( s + i ) − 1 ( i + t + R ) − 1 = a ( s + i ) − 1 ( i + t ) − 1 (1 − R ( t + i ) − 1 ) , which is a compact op er ator.  BIV ARIANT K -THEOR Y AND CORRESPONDENCES 59 W e no w show that the pr o duct of cy cles is a cycle. Note that this result is a generaliza tion of the stability prop er ty of spec tral tr iples pr ov ed in [10]. There it was shown that tenso ring a given s pe c tral triple by a ﬁnitely gener ated pro jective mo dule yields aga in a sp ectr al triple. Lemma 6.3.2. L et s, t b e a p air of almost antic ommu ting selfadj oint r e gular op er- ators in a C k -mo dule E k , such tha t s + t is selfadjo int and r e gular i n E k . If a ( s + i ) − 1 ( t + i ) − 1 , a ( s − i ) − 1 ( t − i ) − 1 ∈ K B k ( E k ) , for some a ∈ End ∗ B k ( E k ) , then a ( s + t ± i ) − 1 ∈ K B k ( E k ) . Pr o of. By a ssumption, a ( s + t ± i ) − 1 ∈ E nd ∗ B k ( E k ), so in view o f corolla r y 5.5.2, it suﬃces to show that a ( s + t ± i ) − 1 ∈ K B ( E ). This in turn is the case if and only if a (1 + ( s + t ) 2 ) − 1 a ∗ ∈ K B ( E ), since for a closed ideal I in a C ∗ -algebra C it is the case that c ∗ c ∈ I ⇔ c ∈ I f or a ll c ∈ C . W e hav e the ide ntities − ia ( s + t + i ) − 1 = a ( s + i ) − 1 ( t + i ) − 1 + a ( s + i ) − 1 ( t + i ) − 1 st ( s + t + i ) − 1 ia ( s + t − i ) − 1 = a ( t − i ) − 1 ( s − i ) − 1 − a ( s − i ) − 1 ( t − i ) − 1 st ( s + t − i ) − 1 , and combining these yields − 2 a (1 + ( s + t ) 2 ) − 1 = a (( s + i ) − 1 ( t + i ) − 1 + ( s + i ) − 1 ( t + i ) − 1 st ( s + t + i ) − 1 + ( t − i ) − 1 ( s − i ) − 1 − ( t − i ) − 1 ( s − i ) − 1 ts ( s + t − i ) − 1 ) . Now use that ( s + i ) − 1 ( t + i ) − 1 + ( t − i ) − 1 ( s − i ) − 1 equals ( t − i ) − 1 ( s − i ) − 1 (2 − [ s, t ])( t + i ) − 1 ( s + i ) − 1 , and ( s + t − i ) − 1 = ( s + t + i ) − 1 + 2 i (1 + ( s + t ) 2 ) − 1 , and write ( s + i ) − 1 = x, ( t + i ) − 1 = y , to ﬁnd − 2 a (1 + ( s + t ) 2 ) − 1 = a ( xy + y ∗ x ∗ + xy [ s, t ]( s + t + i ) − 1 + 2 i y ∗ x ∗ ts (1 + ( s + t ) 2 ) − 1 − x ∗ y ∗ (2 + [ s, t ]) y xts ( s + t + i ) − 1 ) . Since all terms o n the rig ht hand side ar e compact, the left hand side is compact. Therefore a (1 + ( s + t ) 2 ) − 1 and also a (1 + ( s + t ) 2 ) − 1 a ∗ ∈ K B ( E ) as des ir ed.  Theorem 6.3.3. L et k ≥ 1 , A, B , C b e C k -algebr as, ( E , S ) a tr ansverse C k K K - cycle for ( A, B ) and ( F , S ) a tr ansverse C k K K -cycle for ( B , C ) . L et ∇ : E k → E k ˜ ⊗ B k Ω 1 ( B k ) b e a tr ansverse C 1 -c onne ction on E . Then the op er ator S ⊗ 1 + 1 ⊗ ∇ T has A k -lo c al ly c omp act r esolvent. That is, f or a ∈ A k we have a ( S ⊗ 1 + 1 ⊗ ∇ T ± i ) − 1 ∈ K B k ( E k ) . Pr o of. By lemma 6.3.2 it suﬃces to show that a ( s + i ) − 1 ( i + t ) − 1 and a ( s − i ) − 1 ( t − i ) − 1 are compac t fo r a ∈ A k . Since the oper ator s + t := S ⊗ 1 + 1 ⊗ ∇ T is C k , we hav e that a ( s + t ± i ) − 1 ∈ E nd ∗ B k ( E k ) . 60 BRAM M ESLAND By co r ollary 5.5 .2 w e ha ve K B k ( E k ) = E nd ∗ B k ( E k ) ∩ K B ( E ) , so it suﬃces to show that a ( s + t ± i ) − 1 ∈ K B ( E ). By lemma 6.3.1, we only hav e to chec k this in c a se ∇ is the Grassma nn connection on H B . Denote b y { e j } j ∈ Z \{ 0 } the s ta ndard ortho normal basis of H B . Note that ( s ± i ) − 1 ( e j ⊗ f ) = ( S ± i ) − 1 e j ⊗ f , ( t ± i ) − 1 ( e j ⊗ f ) = e j ⊗ ( T ± i ) − 1 f . Cho ose a coun table, increasing, contractive approximate unit for B , suc h that for all 1 ≤ | j | ≤ n ≤ m we hav e k e j ( u n − u m ) k ≤ 1 n . The sequence x n = X 1 ≤| j |≤ n a ( S + i ) − 1 e j ⊗ u n ( i + T ) − 1 ⊗ e j ∈ K C ( H B ˜ ⊗ F ) ∼ = H B ˜ ⊗ K C ( F ) ˜ ⊗ H ∗ B , conv er ges point wis e to a ( s + i ) − 1 ( i + t ) − 1 . W e show it co nv erg es in norm. W e have x m − x n = X 1 ≤| j |≤ n a ( S + i ) − 1 e j ( u m − u n ) ⊗ ( i + T ) − 1 ⊗ e j + X n +1 ≤| j |≤ m a ( S + i ) − 1 e j ⊗ u m ( i + T ) − 1 ⊗ e j . A co mputation in the linking alg ebra yields X 1 ≤| j |≤ n a ( S + i ) − 1 e j ( u m − u n ) ˜ ⊗ ( a ( S + i ) − 1 e j ( u m − u n )) ∗ ≤ k a k 2 2 n n 2 = k a k 2 2 n , and therefor e k X 1 ≤| j |≤ n a ( S + i ) − 1 e j ( u m − u n ) ⊗ ( i + T ) − 1 ⊗ e j k 2 ≤ k a k 2 2 n → 0 . F o r the tail X n +1 ≤| j |≤ m a ( S + i ) − 1 e j ⊗ u m ( i + T ) − 1 ⊗ e j , it is enough to o bserve that k u m ( i + T ) − 1 k ≤ 1 and k X n +1 ≤| j |≤ m a ( S + i ) − 1 e j k → 0 , bec ause a ( S + i ) − 1 is compa ct.  Recall that Ψ 0 ( A, B ) denotes the set of unbounded K K -cyc les up to unitary equiv alence. F o r k ≥ 1, we denote by Ψ k 0 ( A, B ) the set of C k K K -cycles with transverse C k -connection on them, up to transverse C k unitary equiv alence. Note that this requires ﬁxing C k -sp ectral triples for A and B . Theorem 6.3.4. F or k ≥ 1 , the diagr am Ψ k 0 ( A, B ) × Ψ k 0 ( B , C ) ( S, T ) 7→ S ⊗ 1 + 1 ⊗ ∇ T ✲ Ψ k 0 ( A, C ) K K 0 ( A, B ) ⊗ K K 0 ( B , C ) b ❄ ⊗ B ✲ K K 0 ( A, C ) b ❄ c ommutes. BIV ARIANT K -THEOR Y AND CORRESPONDENCES 61 Pr o of. W e just need to chec k that the KK- c y cles ( E , S ), ( F , T ) and ( E ˜ ⊗ B F , S ˜ ⊗ 1+ 1 ⊗ ∇ T ) satisfy the c o nditions o f theorem 2 .2.3. If we wr ite D for S ⊗ 1 + 1 ⊗ ∇ T = s + t , we hav e to c heck that J :=  D 0 0 T  ,  0 T e T ∗ e 0  is b ounded on Do m ( D ⊕ T ). This is a stra ig htf orward calcula tion: J  e ′ ⊗ f ′ f  =  S e ⊗ f + ( − 1) ∂ e ∇ T ( e ) f h e, S e ′ i f + [ T , h e, e ′ i ] f + ( − 1 ) − ∂ e ′ h e, ∇ T ( e ′ ) i f  =  S e ⊗ f + ( − 1 ) ∂ e ∇ T ( e ) f h S e, e ′ i f + h∇ T ( e ) , e ′ i f  . This is v alid whenever e ∈ Dom S ∩ E 1 , which is dense in E . The second condition Dom ( D ) ⊂ Dom ( S ˜ ⊗ 1) is obvious, s o w e turn the semib ound- edness co ndition (6.32) h S ˜ ⊗ 1 x, D x i + h D x, S ˜ ⊗ 1 x i ≥ κ h x, x i , m ust hold for all x in the domain. On Im ( s + λi ) − 1 ( t + λi ) − 1 , w hich is a common core for s and D , the expression 6 . 3 2 is equal to h [ D , S ˜ ⊗ 1] x, x i = h [ s + t, s ] x, x i = h sx, sx i + h [ s, t ] x, x i ≥ −k [ s , t ] kh x, x i , and the last e stimate is v alid since [ s, t ] is in End ∗ C ( E ˜ ⊗ B F ). Thus, it holds for all x in the domain.  The functor K K forgets all the smo othness assumptions impo sed on the cycles in Ψ k 0 ( A, B ). The problem of smo o thening giv e n cycles and equipping them with a connection shall be dealt with els ewhere. Also note that for forming un bo unded Kaspar ov pro ducts, it is enough to hav e a C 1 -cycle with tr ansverse C 1 -connection. It is relev ant for the categ orical consider ations of the next section. 6.4. A category of sp ectral triples. Let A and B be smo o th C ∗ -algebra s. W e saw tha t triples ( E , D, ∇ ) co ns isting of a smo oth ( A, B )-bimo dule eq uipped with a smo oth regula r o pe r ator D and a smo oth connection ∇ form a category , in which the co mpo sition law is ( E k , S, ∇ ) ◦ ( F k , T , ∇ ′ ) := ( E k ˜ ⊗ B k F k , S ⊗ 1 + 1 ⊗ ∇ T , 1 ⊗ ∇ ∇ ′ ) . This can be natur a lly interpreted as a c a tegory o f sp ectral triples. Deﬁnition 6.4.1 . Let A and B b e C ∗ -algebra s, and ( H , D ) and ( H ′ , D ′ ) b e C k sp ectral triples f or A and B resp ectively , with k ≥ 1. A C k - c orr esp ondenc e ( E , S, ∇ ) b etw e e n ( H , D ) and ( H ′ , D ′ ) is a class [( E k , S, ∇ )] ∈ Ψ k 0 ( A, B ) of a C k -( A, B )-bimo dule w ith tr ansverse C k -connection, such that ther e is a unitary isomorphism o f s pec tral tr iples ( H , D ) ∼ = ( E ˜ ⊗ B H ′ , S ⊗ 1 + 1 ⊗ ∇ D ′ ) a nd G ( D i ) ∼ = G ( S ˜ ⊗ 1 + 1 ˜ ⊗ ∇ D ′ ) i for i = 0 , ..., k under this isomorphism. The cor r esp ondence is smo oth if it is C k for a ll k . Two corre s po ndences are said to be equiv alent if they ar e C k - or smo othly unitar ily iso morphic such that the unitary in tertwines the op erator s a nd connectio ns and induces isomor phis ms on the graphs a nd the smo oth structure up to degre e k . The s et of isomorphism clas ses of suc h corresp ondences is denoted by Cor k ( D , D ′ ) o r Co r ( D, D ′ ) in the smo oth ca se. W e can reformulate the previous res ults as a ca tegorical s tatement. 62 BRAM M ESLAND Theorem 6.4. 2. F or k ≥ 1 , ther e is a c ate gory whose obje ct s ar e C k -sp e ctr al triples and whose morphisms ar e the sets Cor k ( D , D ′ ) . The b oun de d tr ansform b ( E , D , ∇ ) = ( E , b ( D )) deﬁnes a funct or Cor k → K K . A catego ry with unbounded C k -cycles as ob jects can b e constructed in a simila r wa y . A morphism of un bo unded cycles A → ( E , D ) ⇌ B and A ′ → ( E ′ , D ′ ) ⇌ B ′ is given by a corre s po ndence A → ( F , S, ∇ ) ⇌ A ′ and a bimo dule B → F ′ ⇌ B ′ , where B is represented by compact op era tors. The bo unded transfo rm functor then takes v alues in the morphism c ate gory K K 2 . F ur thermore, we w ould like to note that the category of spec tr al triples con- structed is a 2- category . A mo rphism of morphisms f : ( E , D , ∇ ) → ( E ′ , D ′ , ∇ ′ ) is given by an element F ∈ Hom ∗ B k ( E k , F k ), inducing mor phisms G ( D i ) i → G ( D ′ i ) i , commuting with the left A i -mo dule structure, in tertwining the co nnections a nd the op erator s. The external pr o duct of corresp o ndences is deﬁned in the exp ected wa y: ( E , D , ∇ ) ⊗ ( E ′ , D ′ ∇ ′ ) := ( E ⊗ E ′ , D ⊗ 1 + 1 ⊗ D ′ , ∇⊗ 1 + 1 ⊗∇ ) . In this wa y , Cor b ecomes a symmetric mo noidal ca tegory . Appendix A. Smoothness and regularity Recall that a sp ectr al triple ( A, H , D ) is r e gular [1 1] if ther e is a dense subalg ebra A ⊂ A such that A and [ D , A ] are in Do m ∞ ad | D | . W e now pro ceed to show that the no tion of smo othness intro duced ab ov e is weak er than r egularity . F o r a reg ular bimo dule we introduce representations π ′ i : A → M 2 i (End ∗ B ( E )) inductively by se tting π ′ 0 ( a ) := a and π ′ i +1 ( a ) :=  π ′ i ( a ) 0 [ | D | , π ′ i ( a )] π ′ i ( a )  . Subsequently , deﬁne repres entations θ ′ i : A → M 2 i (End ∗ B ( E )) b y θ ′ i ( a ) := p | D | [ i ] π ′ i ( a ) p | D | [ i ] + v [ i ] p | D | [ i ] v ∗ [ i ] γ i π ′ i ( a ) γ i v [ i ] p | D | [ i ] v ∗ [ i ] . Here we use γ to deno te the usual diag onal g rading on L 2 i j =0 E . Notice that for i even, the γ ’s disapp ea r fro m the formula. Both the π ′ i and θ ′ i are gr aded representations for the diagonal gr ading, i.e. π ′ i ( ˆ γ ( a )) = γ π ′ i ( a ) γ , because | D | is even. Lemma A.3. Le t E b e an ( A, B ) -bimo dule, D a selfadjoint re gular op er ator in E . F or al l i ther e exist unitaries u i such that u i θ D i u ∗ i = θ ′ i . In p articular Dom θ D i = Do m θ ′ i . Pr o of. The op e rator U i :=  (1 + | D | D ) r ( D ) 2 ( D − | D | ) r ( D ) 2 ( | D | − D ) r ( D ) 2 (1 + | D | D ) r ( D ) 2  ∈ M 2 i +1 (End ∗ B ( E )) , BIV ARIANT K -THEOR Y AND CORRESPONDENCES 63 is unitar y and maps G ( D ) to G ( | D | ). Mo reov er it co mm utes with both D and v i and intert wines the W o ronowicz pro jections: (A.1) U i p D i U ∗ i = p | D | i . Set u 1 := U 1 , and inductively deﬁne u i +1 :=  u i 0 0 u i  U i , so that u i +1 p D [ i +1] u ∗ i +1 = p | D | [ i +1] . The u i int ertwine the θ i ’s: (A.2) u i θ D i ( a ) u ∗ i = θ ′ i ( a ) . T o see this, note tha t θ i = p [ i ] π i p [ i ] + v [ i ] p [ i ] v ∗ [ i ] π i v [ i ] p [ i ] v ∗ [ i ] , and that it is c le ar that up D π D 1 p D u ∗ = p | D | π | D | 1 p | D | . Then up D ⊥ π D 1 ( a ) p D ⊥ u ∗ = uv p D v ∗ π D 1 ( a ) v p D v ∗ u ∗ = uv p D γ 1 π D 1 ( a ∗ ) ∗ γ 1 p D v ∗ u ∗ = uv p D π D 1 ( ˆ γ ( a ∗ )) ∗ p D v ∗ u ∗ = v p | D | π | D | 1 ( ˆ γ ( a ∗ )) ∗ p | D | v ∗ = p | D | ⊥ γ π | D | 1 ( a ) γ p | D | ⊥ . So A.2 holds for i = 1. Supp ose tha t A.2 holds for i . Then since U i +1 p D i +1 p D [ i ] π D i +1 ( a ) p D i +1 p D i U ∗ i +1 = p | D | i +1 p D [ i ]  θ i ( a ) 0 [ | D | , θ i ( a )] θ i ( a )  p D [ i ] p | D | i +1 , and p | D | i +1 commutes with  u i 0 0 u i  , it follows that u i +1 p D [ i +1] π D i +1 ( a ) p D [ i +1] u ∗ i +1 = p | D | i +1 u i p D [ i ] θ i ( a ) p D [ i ] u ∗ i 0 [ | D | , u i p D [ i ] θ i ( a ) p D [ i ] u ∗ i ] u i p D [ i ] θ i p D [ i ] ( a ) u ∗ i ! p | D | i +1 = p | D | i +1 p | D | [ i ]  θ ′ i ( a ) 0 [ | D | , θ ′ i ] θ ′ i ( a )  p | D | [ i ] p | D | i +1 = p | D | [ i +1] π ′ i +1 ( a ) p | D | [ i +1] . Using either 4.15 or 4.16 and the fact that u i and v [ i ] commute, one o btains that u i +1 v [ i +1] p D [ i +1] v ∗ [ i +1] π D i +1 ( a ) v [ i +1] p D [ i +1] v ∗ [ i +1] u ∗ i +1 = v [ i +1] p | D | i +1 v ∗ [ i +1] γ i π ′ i ( a ) γ i v [ i +1] p | D | i +1 v ∗ [ i +1] , in the same way a s for i = 1 . Th us, u i +1 θ D i +1 u ∗ i +1 = θ ′ i +1 .  Theorem A. 4. L et ( E , D ) b e a r e gular unb ou n de d ( A, B ) -bimo dule. Then ( E , D ) is smo oth. 64 BRAM M ESLAND Pr o of. W e will show that A ⊂ A n for all n . By deﬁnition, A ⊂ A 1 , so supp ose A ⊂ A n . Then θ n ( a ) is w ell deﬁned, and we hav e to show that [ D , θ n ( a )] extends to an adjointable o p erator. F r om lemma A.3 it follows that for a ∈ A , [ D , θ n ( a )] = u n [ D , θ ′ n ( a )] u ∗ n = u n ( p [ n ] [ D , π ′ n ( a )] p [ n ] + v [ n ] p [ n ] v ∗ [ n ] [ D , γ n π ′ n ( a ) γ n ] v [ n ] p [ n ] v ∗ [ n ] ) u ∗ n = u n ( p [ n ] [ D , π ′ n ( a )] p [ n ] + ( − 1) n v [ n ] p [ n ] v ∗ [ n ] γ n [ D , π ′ n ( a )] γ n v [ n ] p [ n ] v ∗ [ n ] ) u ∗ n . Since ( E , D ) is regular , [ D , A ] ⊂ Dom (ad | D | ) n , which is the same a s saying that (ad | D | ) n ( A ) ⊂ Dom (ad D ) . Therefore w e hav e that [ D , π ′ n ( a )] ∈ M 2 n (End ∗ B ( E )) for a ∈ A . It follo ws that A ⊂ A n +1 as des ir ed.  Appendix B. No nunit al C k -algebras In t his app endix we describe a principle t o reduce the deﬁning representation of a C k algebra in Sob k ( D ) to the ess e ntial cas e. Note that a homo morphism π : A → B betw een op era tor a lgebras is essen tial when π ( A ) B π ( A ) is dense in B . In that case π extends to a map M ( A ) → M ( B ) (see [8]). When B is unital and A is cb-isomor phic to π ( A ), w e hav e M ( A ) ∼ = { T ∈ B : T π ( A ) , π ( A ) T ⊂ π ( A ) } . In what follo ws , w e will use some well known prop e rties of the strong a nd w eak top ologies on the algebra of b ounded o pe r ators on a s eparable inﬁnite dimensiona l Hilber t s pace. Recall that, for a Hilbert space H , the str ong op er ator top olo gy on B ( H ) is the top ology o f p o in t wise no rm co nvergence: T i → T strongly if T i ξ → T ξ in no rm for all ξ ∈ H . The we ak op er ator top olo gy is the weakest top olog y that makes the functionals T 7→ h T ξ , η i c ontin uous, so T i → T weakly when h T i ξ , η i → h T ξ , η i for all ξ , η ∈ H . These top olog ies are complete on the closed unit ball of B ( H ). Hence a sequence T i with sup i k T i k ≤ C that is strong ly or weakly Cauc h y has a limit in B ( H ) in the resp ective topolo gies. Moreover, ope r ator multiplication is se parately contin uo us for these topologie s : if T i → T strong ly or w eakly , then S T i → S T , T i S → T S strongly o r weakly , resp ectively . Lemma B. 5. L et a n , b ∈ Sob i ( D ) b e a se quenc e such that π i ( a n ) → π i ( b ) str ongly r esp. we akly. Then θ i ( a n ) → θ i ( b ) str ongly r esp. we akly. Pr o of. W e prove the statement for the weak topolo gy: According to (4.13) we ha ve h θ i ( a n ) ξ , η i = h p i p i − 1 π i ( a n ) p i − 1 p i ξ , η i + h p ⊥ i p ⊥ i − 1 π i ( a n ) p i − 1 p i ξ , η i , from which the statement is immediate.  Prop ositi o n B.6 . L et B b e a nonunital C k algebr a with sp e ctr al triple ( H , D ) . L et p b e the pr oje ction onto the essential su bsp ac e B H . Then p ∈ Sob k ( D ) and π k ( b ) = π k ( p ) π k ( a ) π k ( p ) . BIV ARIANT K -THEOR Y AND CORRESPONDENCES 65 Pr o of. When B is unital ther e is nothing to pr ov e. A nonunital C k -algebra B by deﬁnition pos esses an even, increas ing , contractive approximate unit u n , which is completely b ounded in B k . Denote b y p ∈ B ( H ) the pro jectio n onto B H , the essential subspace o f B . It is well known that u n → p in the weak op erator top olo g y on H . Since u n (1 − p ) H = u n ( B H ) ⊥ = 0, and u n b → b in norm, it follows that ( u n − p ) h = ( u n − p )( ph + (1 − p ) h ) = ( u n − p ) ph → 0 , that is, u n → p strongly . W e will show that p ∈ Sob k ( D ) by induct ion on k . Supp ose that for k − 1 w e hav e shown that p ∈ Sob k − 1 ( D ) and π k − 1 ( u n ) → π k − 1 ( p ) stro ng ly . By lemma B.5 we get that θ k − 1 ( u n ) → θ k − 1 ( p ) stro ngly . Then, we need to show that θ k − 1 ( p ) Dom D ⊂ Dom D and [ D, θ k − 1 ( p )] is b ounded on Dom D . T o this end ob- serve that [ D , θ k − 1 ( u n )] is a uniformly b ounded sequence of o per ators that is weakly Cauch y: h [ D , θ k − 1 ( u n )] ξ , η i = h θ k − 1 ( u n ) ξ , D η i − h θ k − 1 ( u n ) D ξ , η i → h θ k − 1 ( p ) ξ , D η i − h θ k − 1 ( p ) D ξ , η i , for ξ , η ∈ Do m D , which is dense. Therefore π k ( u n ) has a w eak limit in B ( L 2 k i =1 H ), which we denote b y q . Denote b y p k the pro jection on to π k ( B k )( L 2 k i =1 H ). Then since π k ( u n ) π k ( b ) → π k ( b ) in norm, we ﬁnd that π k ( u n ) → 1 strongly on Im p k , and hence q p k = p k . On the other hand, taking weak limits w e ﬁnd p k q = lim n p k π k ( u n ) = lim n π k ( u n ) = q , and so Im p k = Im q = π k ( B k )( 2 k M i =1 H ) . Therefore q is an idempo tent: q 2 h = lim π k ( u n ) q h = q h. Applying the ab ov e pr o cedure to π k ( u n ) ∗ yields Im q ∗ = π k ( B k ) ∗ ( L 2 k i =1 H ) and th us Im (1 − q ) = ( Im q ∗ ) ⊥ = π k ( B k ) ∗ ( 2 k M i =1 H ) ⊥ , from which we see that h π k ( b )(1 − q ) ξ , η i = h (1 − q ) ξ , π k ( b ) ∗ η i = 0 , for a ll ξ , η s o π k ( b )(1 − q ) = 0. Consequently π k ( u n ) ξ = π k ( u n ) q ξ → q ξ , i.e. π k ( u n ) → q strongly . T aking ξ =  η D η  , we ﬁnd π k ( u n )  η D η  =  θ k − 1 ( u n ) η D θ k − 1 ( u n ) η  →  θ k ( p ) η D θ k ( p ) η  , so θ k − 1 ( p ) pre s erves Dom D and [ D , θ k − 1 ( p )] is a densely de ﬁned op era tor that is the weak limit of t he bounded sequence [ D , θ k − 1 ( u n )], so it m ust be bounded. Hence 66 BRAM M ESLAND q = π k ( p ) and π k ( u n ) → π k ( p ) in t he stro ng topo logy , completing the induction step.  Theorem B.7 . The inclusion B k ⊂ Sob k ( D ) natur al ly extends to an inclusion M ( B k ) ⊂ Sob k ( D ) of involutive op er ator algebr as. In p articular M ( B k ) is sp e ctr al invariant in its C ∗ -closur e. Pr o of. Let p be the pro jection on the essential subs pace B H . The inclusion B k ⊂ Sob k ( D ) co n tracts to an inclusion B k ⊂ π k ( p )Sob k ( D ) π k ( p ). This inclusion is essential, and the op erator algebra π k ( p )Sob k ( D ) π k ( p ) is unital. Ther efore M ( B k ) = { a ∈ π k ( p )Sob k ( D ) π k ( p ) : ab, ba ∈ B k } , which is by deﬁnition a subalge br a of So b k ( D ).  Theorem B. 8. L et E ⇌ B b e a C k -mo dule over the C k -algebr a B , with sp e ctr al tiple ( H , D ) . The m ap End ∗ B k ( E k ) → k M i =0 B ( E k ˜ ⊗ B k 2 i M j =1 H ) T 7→ T ⊗ π [ k ] ( p ) , is a cb-isomorphism onto its image. The same h olds for K B k ( E k ) Pr o of. By prop osition B.7, the pro jectio n p ca n b e used to turn π [ k ] : B k → k M i =0 B ( 2 i M j =1 H ) , (cf.(4.14)) into a n essential repr e sentation on the Hilb ert space π [ k ] ( p ) k M i =0 ( 2 i M j =1 H ) . 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Anal., 76 (1988), 217-230. [28] S.L. W orono wicz, Unb ounde d element s aﬃliate d with C ∗ -algebr as and non-c omp act q uantum gr oups , Commun.Math.Ph ys. 136, 399-432. Ma themati sch Instituut, Universiteit Ut recht, Budapest laan 6, 3584 CD Ut recht, The Netherlands E-mail addr ess : brammesland@gmai l.com

Unbounded bivariant $K$-theory and correspondences in noncommutative geometry

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