Banach spaces with projectional skeletons

Banac h spaces with pro jectional sk eletons Wies la w Kubi ´ s ∗ Institute of Mathematic s, J an Kochano wski Univ ersit y in Kielce, P oland and Mathematica l Institute of the Czec h Academ y of Sciences, Pr ague, Czech Repub lic Octob er 26, 2018 Abstract A pro jectional sk eleton in a Banac h space is a σ -d ir ected family of pro jections on to separable subspaces, co v ering the en tire space. T he class of Banac h spaces with p ro jectional skele tons is strictly larger than the class of P lichk o sp aces (i.e. Banac h spaces w ith a coun tably norming Marku s hevic h basis). W e sho w th at ev- ery space with a pro jectional sk eleton has a pro jectional resolution of the iden tit y and has a normin g space with similar pr op erties to Σ-spaces. W e c haracterize the existence of a pro jectional sk eleton in terms of elemen tary substr uctures, p ro vid- ing simple pro ofs of kno wn results concerning wea kly compactly generated spaces and P lic hk o spaces. W e pro v e a preserv atio n result for Plic hko Banac h sp aces, in v olving trans finite sequences of pr o jectio ns. As a corollary , we sho w that a Banac h s pace is Plic hk o if and only if it h as a comm utativ e pro jectional sk eleton. Mathematics Sub ject C la ssification (2000): 46B26, 46B03, 46E15, 54C35. Keyw ords: Pro jection, pro jectional sk eleton, norming set, pro jectiv e sequence, Plic hk o space. ∗ Research suppo rted by MNiSW gr ant No. N20 1 024 3 2/090 4. Author’s e-mail addr ess: wkubis @pu.k ielce.pl 1 Con ten ts 1 In tro duction 2 2 Preliminaries 4 3 Elemen tary submo dels and pro jections 5 3.1 Pro jections induced b y elemen tary substructures . . . . . . . . . . . . . 9 3.2 W CG spaces and Plichk o pairs . . . . . . . . . . . . . . . . . . . . . . . 10 3.3 Pro jectional generator s . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.4 Bandlo w’s Property Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4 Pro jectional sk eletons 13 4.1 Definition and basic pr o p erties . . . . . . . . . . . . . . . . . . . . . . . 13 4.2 Pro jectional resolutions of the iden tit y . . . . . . . . . . . . . . . . . . 16 4.3 Norming space induced by a pro jectional sk eleton . . . . . . . . . . . . 16 5 Plic hk o spaces and pro jectional skeletons 19 5.1 Preserv ation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.2 A characterization of Plic hko spaces . . . . . . . . . . . . . . . . . . . . 23 6 Spaces of contin uous functions 23 7 Final remarks and op en problems 27 1 In tro d uction It is w ell kno wn that a Bana ch space in whic h ev ery closed subspace is complemen ted is nec essarily isomorphic to a Hilb ert space (Lindenstrauss & Tzafriri [30]). On the other hand, there exist Banac h spaces in which only finite-dimensional and co-finite- dimensional subspaces are complemen t ed (Go w ers & Maurey [11]). There eve n ex- ist (necessarily , non-separable) C ( K ) spaces with no nontrivial b ounded pro jections (Koszmider [21] and Plebanek [33]). In the p ositiv e direction, one has to men tion the w ork of Heinric h & Mankiewicz [15] where, using substructures of ultrap ow ers of Banac h spaces, the authors sho w in par t icular the existence of non-trivial b ounded pro jections in ev ery dual Banach space of densit y greater than the con tin uum (see [38] for an elemen tary pro of ). 2 W e are intere sted in Banac h spaces whic h ha v e “many” pro jections on to separable subspaces. P erhaps one of the most general classe s of this sort w ould b e the class of Banac h spaces with the sep ar able c omplementation pr o p erty (SCP): a space X has the SCP if, b y definition, for ev ery countable set S ⊆ X there is a b ounded pro jection P : X → X suc h that im P is separable a nd S ⊆ im P . It turns out that this pr o p ert y is general enough in order to include somewhat pathological spaces. F or a surv ey on the complemen tation prop erty and its v a r iations we refer to [37]. Another p ossible class of Ba nac h spaces with many pro jections are spaces with the pr oj e ctional r esolution of the identity (PRI), notion in tr o duced b y Linden strauss [28, 29], defined to b e a w ell ordered contin uo us chain of pro jections on to smaller subspaces . This prop erty to g ether with transfinite induction allow s proving v ario us prop erties o f a non- separable Banach space, e.g. a locally uniformly r o tund/Kadec renorming [4 2, 6] a nd the existence o f a linear injection in to c 0 (Γ) [1, 4 1]. See [10 ] or [7 ] f o r more informat io n concerning the PRI metho d. Unfortunately , the existenc e of a PRI in a Banac h space is go o d enough only for densit y ℵ 1 , o therwise it do es no t ev en imply the SCP . W e prop o se a natural class of Banac h spaces whic h ha v e a family of pro jections on to separable subs paces, indexed b y a σ -directed pa r t ially ordered set and satisfying some natural conditions, similar to a PRI. W e call it a “pro jectional sk eleton”. It turns out that a Banac h space with a pro jectional sk eleton lo oks “ almost” like a Plichk o space, i.e. a Banac h space with a countably norming Mar kushevic h ba sis (see [34, 35, 3 6] and [40]). In fact, w e ess en tially kno w only one basic example distinguishing those t w o classes: the space C ( K ), where K is the ordina l ω 2 + 1, endo w ed with the interv al top olog y . This w as sho wn b y Kalenda in [20]. Banac h space s with a pro jectional sk eleton can b e c haracterized by a prop ert y inv o lving norming sets and coun table elemen tary submo d- els. W e explain ho w to use elemen tary s ubmo dels of (some initial part of ) set theory for constructing b ounded pro jections and w e show tha t the existenc e of a pro jectional sk eleton is equiv alen t to some natura l mo del-theoretic prop erty of a suitable norming space, similar to t he existence of the so-called p r oje ctional gener ator . As an application, w e get short pro ofs o f some w ell known results, lik e the existence of a PRI in ev ery w eakly compactly generated space. Our c haracterization allo ws us to sho w that the class of Banac h spaces with a pro jectional sk eleton of norm o ne is stable under arbi- trary c 0 - a nd ℓ p -sums (1 6 p < ∞ ) . W e apply elemen tary submo dels fo r constructing a PRI fro m a pro jectional sk eleton. The main part (Section 5) is dev oted to Plic hk o spaces. W e pro v e a preserv ation result for inductive limits of certain pro jectiv e sequences of Plic hko spaces, similar in spirit to Gul ′ k o’s results o n subspaces of the Σ-pro ducts [12, 13, 14]. As a n application, w e sho w that a Banach space is Plichk o if and only if it has a comm utativ e pro j ectional sk eleton. 3 Finally , we discuss retractional sk eletons in compact spaces – a notion dual to pro jec- tional ske leton. W e c haracterize this class of compacta b y means of elemen tary sub- mo dels and w e state a preserv ation pro p ert y for V aldivia compacta, dual to the cor- resp onding result for Banach spaces. Retractional sk eletons we re in tro duced alr eady in [25], where it is prov ed that V a ldivia compacta are precis ely those compact space s whic h hav e a commutativ e retractional sk eleton. F or more information a nd recen t re- sults concerning V aldivia compacta and their spaces of con tin uous functions w e refer t o [18, 17, 25, 2 6, 3, 23]. 2 Preliminaries W e shall consider Banac h spaces o v er the field of real n um bers, although the results are true also for the complex case. Belo w w e recall most relev a n t notions, definitions and notation. By a pr oje c tion in a Banach space X w e mean a b ounded linear op erator P : X → X suc h that P ◦ P = P . In t his case im P = { x ∈ X : x = P x } and k er P = { x − P x : x ∈ X } = im(id X − P ), where id X is the iden tit y map. Recall tha t a space X has the sep ar able c omplementation pr op erty ( SC P for s hort) if for ev ery countable set S ⊆ X there is a pro jection P : X → X suc h that P X is a separable space containing S . Giv en B ⊆ X ∗ , we write ⊥ ( B ) = { x ∈ X : ( ∀ b ∈ B ) b ( x ) = 0 } . The righ t annihilator ( A ) ⊥ is defined similarly . Let λ be a limit ordinal. A pr oje ctional se quenc e of length λ in a Banac h space X is a sequence of pro jections { P ξ } ξ <λ satisfying the follow ing conditions: 1. ξ < η = ⇒ P ξ = P η ◦ P ξ = P ξ ◦ P η , 2. P δ X = cl( S ξ <δ P ξ X ) for ev ery limit ordinal δ < λ , 3. X = cl ( S ξ <λ P ξ X ). A s p ecial case is a pr oje ctional r esolution of the identity ( PRI ): this is a pro j ectional sequence { P ξ } ξ <λ suc h that k P ξ k = 1 a nd dens( P ξ X ) 6 | ξ | + ℵ 0 for eac h ξ < λ , where | ξ | denotes the cardinality of ξ . All top o lo gical spaces are assumed to b e completely r egula r . T he closure of a set A in a space X will b e denoted b y cl( A ) or, more precisely , b y cl X ( A ). If X is a dual to a Banac h space then cl ∗ will denote the w eak ∗ closure, i.e. the closure with r esp ect to the w eak ∗ top ology on X . A space X is c ountably tight if for ev ery A ⊆ X and for eve ry p ∈ cl A there exists A 0 ∈ [ A ] ℵ 0 with p ∈ cl A 0 . Let Γ b e a set. Giv en x ∈ R Γ , we denote b y suppt( x ) the supp ort 4 of x , i.e. suppt( x ) = { α ∈ Γ : x ( α ) 6 = 0 } . The set Σ(Γ) = { x ∈ R Γ : | suppt( x ) | 6 ℵ 0 } is called a Σ -pr o duct . A V aldivia c om p a c t is a compact space homeomorphic to K ⊆ [0 , 1] κ satisfying K = cl( K ∩ Σ( κ )). A C orson c omp act is, b y definition, a compact subset of Σ( κ ) for some κ . Give n a compact K , D ⊆ K is called a Σ -subset of K if there is a ho meomorphic em b edding h : K → [0 , 1] κ suc h t ha t D = h − 1 [Σ( κ )]. Let h X, k · ki b e a Ba nac h space. A set D ⊆ X ∗ is norm i n g if the form ula (*) | x | = sup {| ϕ ( x ) | / k ϕ k : ϕ ∈ D \ { 0 }} defines an equiv alen t norm o n X . More precisely , we sa y that D is r -norming if k x k 6 r | x | for ev ery x ∈ X . D is 1 -norming if | · | = k · k . The follo wing fact is we ll kno wn. Prop osition 2.1. L et D b e a norming subset of X ∗ . Then D is 1 - norming with r esp e ct to the norm | · | define d by e quation (*). Pr o of. Let D ′ b e the linear span of D and let D 1 = { ϕ ∈ D ′ : k ϕ k 6 1 } . Then | x | = sup ϕ ∈ D 1 | ϕ ( x ) | . Th us, it remains to notice that D 1 = { ϕ ∈ D ′ : | ϕ | 6 1 } . Indeed, if ϕ ∈ D 1 then | ϕ ( x ) | 6 1 whenev er | x | 6 1 , so | ϕ | 6 1. Conv ersely , if ϕ ∈ D \ D 1 then there is x 0 ∈ X suc h that | x 0 | = 1 and | ϕ ( x 0 ) | > 1, so | ϕ | > 1. Recall that a Ba nac h space X is called Pli c hko if there exists a one- t o -one w eak ∗ con tin- uous linear op era t o r T : X ∗ → R κ suc h that T − 1 [Σ( κ )] is norming for X . Equiv alen tly: there are a linearly dense set A ⊆ X and a norming set D ⊆ X ∗ suc h that f or ev ery y ∈ D the set { a ∈ A : y ( a ) 6 = 0 } is countable. If additionally D is linear, w e shall say that h X , D i is a Plichko p air . More generally , we sa y tha t h Y , D i is a Plic hk o pair in a space X if Y is a closed linear subspace of X and h Y , D ′ i is a Plic hk o pa ir, where D ′ = { y ↾ Y : y ∈ D } . Giv en A ⊆ X , the set suppt ( y , A ) = { a ∈ A : y ( a ) 6 = 0 } will b e called the A -supp ort of y ∈ X ∗ . The space D = { y ∈ X ∗ : | suppt( y , A ) | 6 ℵ 0 } is called a Σ -sp ac e . A particularly in teresting sub class of Plic hk o spaces is the class of w eakly Lindel¨ of determined spaces, in tro duced b y V aldivia [39]. A Ba nac h space X is we akly Lindel¨ of determine d ( WLD for short) if h X , X ∗ i is a Plic hk o pair, i.e. X ∗ is a Σ-space. This is equiv alen t to saying that the dual unit ball with the w eak ∗ top ology is Corson compact (see [3 1, Prop. 4.1]). 3 Elemen tary s ubmo dels and pr o je ctions In this section we in tro duce the metho d of elemen tary submo dels, which will b e used extensiv ely throughout the pap er. In the contex t o f retractio ns – top ological counte r- parts of linear pro jections – elemen ta ry submodels turned out to b e a n imp orta nt to ol 5 in [23, 25]. W e refer to the surv ey article of D o w [8], where sev eral applications of elemen tary submodels in set-theoretic top o logy are ex plained. Let N b e a fixed set. The pair h N , ∈i , where ∈ is restricted to N × N , is a structure of the lang ua g e of set theory . Giv en a formula ϕ ( x 1 , . . . , x n ) with all free v aria bles sho wn and giv en a 1 , . . . , a n ∈ N one defines the relation “ h N , ∈i satisfies ϕ ( a 1 , . . . , a n )” (briefly “ h N , ∈i | = ϕ ( a 1 , . . . , a n )” or just “ N | = ϕ ( a 1 , . . . , a n )”) in the usual w a y , by induction on the length of the formula. Namely , N | = a 1 ∈ a 2 iff a 1 ∈ a 2 and N | = a 1 = a 2 iff a 1 = a 2 . It is clear how “satisfaction” is defined fo r conjunction, disjunction and negation. Finally , if ϕ is of the form ( ∃ y ) ψ ( x 1 , . . . , x n , y ) then N | = ϕ ( a 1 , . . . , a n ) iff there exists b ∈ N suc h that N | = ψ ( a 1 , . . . , a n , b ). As an example, if s = { a, b, c } and s, a, b ∈ N while c / ∈ N , then N satisfies “ s has at most tw o elemen ts”, b ecause f or ev ery x ∈ N if x ∈ s then either x = a or x = b . Instead of t he ab ov e definition, some authors use relativization, see e.g. K unen’s b o ok [27]. Giv en a form ula ϕ , the r e l a tivi z a tion of ϕ to N is a form ula ϕ N whic h is built from ϕ b y replacing each quan tifier of the form “ ∀ x ” b y “ ∀ x ∈ N ” and eac h quantifier of the form “ ∃ x ” b y “ ∃ x ∈ N ” . By this w a y , N | = ϕ ( a 1 , . . . , a n ) iff ϕ N ( a 1 , . . . , a n ) ho lds (of course, a 1 , . . . , a n m ust b e elemen ts of N ). Giv en a set x , w e define the tr ansitive closur e of x to b e tc( x ) = S n ∈ ω tc n ( x ), wh ere tc 1 ( x ) = x ∪ S x and tc n +1 ( x ) = t c 1 (tc n ( x )). In other w ords: y ∈ tc( x ) iff there are x 0 ∈ x 1 ∈ · · · ∈ x k suc h that y ∈ x 0 and x k ∈ x . Thanks to the Axiom of Regularity , these tw o definitions of tr ansitiv e closure are equiv alen t and ev ery set of the form tc( x ) is tr ansitive , i.e. y ∈ t c( x ) implies y ⊆ tc( x ). Giv en a cardinal θ , w e denote by H ( θ ) the class of all sets whose tr ansitiv e closure has cardinalit y < θ . It is w ell kno wn that H ( θ ) is a set, not a prop er class. It is clear that H ( θ ) is transitiv e. W e shall consider elemen tary subs tructures of h H ( θ ) , ∈i . It is w ell kno wn that for a regular uncoun table cardinal θ , the structure h H ( θ ) , ∈i satisfies all the axioms of set theory , exce pt p ossibly the P o w er Set Axiom, see [2 7, IV.3]. Recall tha t a substructure M of h H ( θ ) , ∈i is called elementary if for ev ery form ula ϕ ( x 1 , . . . , x n ) with all free v ariables sho wn, for ev ery a 1 , . . . , a n ∈ M , w e hav e that M | = ϕ ( a 1 , . . . , a n ) ⇐ ⇒ H ( θ ) | = ϕ ( a 1 , . . . , a n ) . The fact that M is an elemen tary submo del of h H ( θ ) , ∈i is denoted b y M  h H ( θ ) , ∈i . The reason for using elemen tary submodels of H ( θ ) is that t hese structures satisfy most of the axioms of set theory: if θ > ℵ 0 is regular then H ( θ ) satisfies all the axioms except p ossibly the p ow er-set, b ecause it ma y happ en that 2 λ > θ f or some λ < θ . Moreov er, in practice it is usually easy to point o ut a cardinal θ suc h that H ( θ ) satisfies giv en finitely many form ulas with parameters, needed for applications. Another useful f eat ure of H ( θ ) with θ regular is the fact for eve ry formula ϕ ( x 1 , . . . , x n ) in whic h all quantifie rs 6 are b ounded (i.e. of the for m “ ∀ x ∈ y ” or “ ∃ x ∈ y ”) a nd for ev ery a 1 , . . . , a n ∈ H ( θ ), ϕ ( a 1 , . . . , a n ) ho lds if and only if H ( χ ) | = ϕ ( a 1 , . . . , a n ). F or more info r ma t ion, see [27, IV.3]. Since in most cases we indeed use form ulas with b ounded quan tifiers, one can simply “chec k” their v alidity b y lo oking at a sufficien tly large H ( θ ). One can also use R efle ction Principle , whic h say s that given a form ula of se t theory ϕ ( x 1 , . . . , x n ) and giv en sets a 1 , . . . , a n suc h that ϕ ( a 1 , . . . , a n ) holds, there exists θ suc h that the structure h H ( θ ) , ∈i satisfies ϕ ( a 1 , . . . , a n ). In some cases θ ma y not b e regular, although it ma y b e arbitrarily big and it ma y ha v e arbitrarily big cofinalit y . More precisely: the class o f cardinals θ with the ab o v e prop erty is closed and unbounded. Th us, when considering finitely man y fo r mulas a nd parameters, w e can “c hec k” their v alidity b y restricting atten tion to H ( θ ), where θ is a “big enough” cardinal, meaning that the cofinality of θ is gr eat er than a prescribed cardinal and a ll relev an t formulas are satisfied in h H ( θ ) , ∈i . Summarizing: assume we would like to use in our argumen ts fo rm ulas ϕ 1 , . . . , ϕ n and parameters from a finite se t S . W e then find a cardinal θ so that S ⊆ H ( θ ) and, b y Reflection Principle, all v a lid form ulas ϕ 1 , . . . , ϕ n with suitable parameters ar e satisfied in H ( θ ). F inally , w e shall use elemen tary substructures of H ( θ ) whic h contain S . If the form ulas ϕ 1 , . . . , ϕ n ha v e only b ounded quan tifiers, then w e do not need to use Reflection Principle, since the form ulas will b e satisfied in ev ery H ( θ ) with θ big enough, i.e. ev ery regular θ greater than some fixed cardinal θ 0 . In o rder to illustrate elemen ta r it y , let us come back to the simple example des crib ed ab ov e: let s = { a, b, c } ∈ N , a, b ∈ N and now assume that N  H ( θ ) and that a, b, c are pairwise distinct. Since N ⊆ H ( θ ), w e see that s ∈ H ( θ ) and consequen tly also c ∈ H ( θ ). Clearly , H ( θ ) | = c ∈ s , therefore H ( θ ) satisfies “there is x ∈ s suc h t ha t neither x = a nor x = b ”. By elemen tarit y , N satisfies the same statemen t, whic h means that there exists d ∈ N suc h that N | = ( d ∈ s ∧ d 6 = a ∧ d 6 = b ). This is a conjunction of atomic form ulas and their negations, so indeed d ∈ s and d / ∈ { a, b } . But w e ha v e assum ed t ha t s ∩ N = { a, b } , whic h is a contradiction. This example sho ws that elemen tary substructures of H ( θ ) “k eep” elemen ts of a finite set. In general, if N  H ( θ ) then s ∈ N do es not necessarily imply that s ⊆ N , unless s is countable or N contains a sufficie n tly big initial inte rv al of ordinals, see Proposition 3.1(c) b elo w or [8, Thm. 1.6]. A particular case of the L¨ owenhe im-Sk olem Theorem (for the languag e of set theory) sa ys that for ev ery infinite set S ⊆ H ( θ ) there exists M  h H ( θ ) , ∈i suc h that | M | = | S | . This theorem can b e view ed as the “ultimate” closing-off ar g umen t and its ty pical pro of indeed pro ceeds b y “closing-off” the giv en set S , by adding elemen ts whic h witness “satisfaction” of all suitable form ulas of the f orm ( ∃ x ) ψ . Imp ortant for applicatio ns is the fact that, thanks to the L¨ o wenhe im-Sk olem theorem, 7 w e ma y consider c ountable elemen tary subs tructures of a n arbitrarily large H ( θ ). Prop osition 3.1. L et θ b e an unc ountable r e gular c ar dinal and let M  h H ( θ ) , ∈i . (a) Assume u ∈ H ( θ ) , a 1 , . . . , a n ∈ M an d ϕ ( y , x 1 , . . . , x n ) is a formula such that u is the unique element of H ( θ ) for which H ( θ ) | = ϕ ( u, a 1 , . . . , a n ) . Then u ∈ M . (b) L et s ⊆ M b e a fin ite set. Then s ∈ M . (c) If S ∈ M is a c ountable set then S ⊆ M . Pr o of. (a) By elemen tarity there exists v ∈ M such that M | = ϕ ( v , a 1 , . . . , a n ). Using elemen tarit y again, w e see that H ( θ ) | = ϕ ( v , a 1 , . . . , a n ). Thus u = v . (b) Let s = { a 1 , . . . , a n } . Then s is the unique set satisfying the formula ϕ ( s, a 1 , . . . , a n ), where ϕ ( x, y 1 , . . . , y n ) is ( ∀ t ) t ∈ x ⇐ ⇒ t = y 1 ∨ t = y 2 ∨ · · · ∨ t = y n . Applying (a), w e see that s ∈ M . (c) By induction and by ( a ), w e see that all natural num b ers are in M . Also by (a), the set of natural n um b ers ω is an elemen t of M , b eing uniquely defined as the minimal infinite ordinal. Notice tha t H ( θ ) satisfies “there exists a surjection from ω on to S ”. By elemen tarit y , t here exists f ∈ M suc h that M satisfies “ f is a surjection from ω on to S ”. Again using (a), w e see that f ( n ) ∈ M for eac h n ∈ ω . Finally , it suffices to observ e tha t f is indeed a surjection, i.e. for ev ery x ∈ S there is n suc h that x = f ( n ). This follows from elemen tarit y , b ecause assuming f [ ω ] 6 = S , the form ula “( ∃ x ∈ S )( ∀ n ∈ ω ) x 6 = f ( n )” w ould b e satisfied in M , con tradicting that f is a surjection. Fix a Bana c h space X and c ho ose a “ big enough” regular cardinal θ , so tha t X ∈ H ( θ ). T ak e an elemen tary substructure M of h H ( θ ) , ∈i suc h that X ∈ M . No t e that M may b e coun table, b y the L¨ ow enheim-Sk olem Theorem. What can w e sa y ab out the set X ∩ M ? By elemen tarit y , it is closed under addition. By Prop osition 3.1(a), the field of rationals is con tained in M , therefore X ∩ M is a Q -linear subspace of X . Consequen tly , the norm closure of X ∩ M is a Banac h subspace of X . In particular, the w eak closure of X ∩ M equals the nor m closure of X ∩ M . W e shall write X M instead of cl ( X ∩ M ) and we shall call X M the subsp ac e induc e d by M . In case of some t ypical Banac h spaces, we can describ e the subspace X M . F or instance, let X = ℓ p (Γ), where 1 6 p < ∞ and Γ is an uncoun table set. Then X M can b e iden tified with ℓ p (Γ ∩ M ). Indeed , iden tify x ∈ ℓ p (Γ ∩ M ) with its extension x ′ ∈ ℓ p (Γ) defined b y x ′ ( α ) = 0 for α ∈ Γ \ M . Let x ∈ X ∩ M . Then suppt( x ) = { α ∈ Γ : x ( α ) 6 = 0 } 8 is a coun table set and hence, by eleme n tarity , it b elong s t o M . By Prop o sition 3.1(c), suppt( x ) ⊆ M . Th us x ∈ ℓ p (Γ ∩ M ). On the other hand, if x ∈ ℓ p (Γ ∩ M ) then a rbitrarily close to x w e can find y ∈ ℓ p (Γ ∩ M ) suc h that s = suppt( y ) is finite. Moreov er, w e ma y assume that y ( α ) ∈ Q for α ∈ s . By Prop osition 3 .1 (b), y ↾ s ∈ M and consequen tly also y ∈ M . Hence x ∈ cl( X ∩ M ) = X M . Giv en a compact space K ∈ H ( θ ) and M  h H ( θ ) , ∈i , define t he follo wing equiv alence relation ∼ M on K : x ∼ M y ⇐ ⇒ ( ∀ f ∈ C ( K ) ∩ M ) f ( x ) = f ( y ) . W e shall write K / M instead of K / ∼ M and w e shall denote by q M (or, more precise ly , q M K ) the canonical quotient map. It is not hard to chec k that K / M is a compact Haus- dorff space of w eigh t not exceeding the cardinality of M . T his construction has been used b y Bandlo w [4, 5] for c haracterizing Corson compacta in terms of elemen tary substructures. Lemma 3.2. L et K b e a c om p a ct sp ac e, let θ b e a big enough r e g ular c ar dinal and let M  h H ( θ ) , ∈i b e s uch that K ∈ M . Then cl( C ( K ) ∩ M ) = { ϕ ◦ q M : ϕ ∈ C ( K / M ) } , wher e cl de n otes the norm closur e in the ab o v e form ula. Pr o of. Let Y denote the set on the right-hand side. Then Y is a closed linear subspace of C ( K ). Given ψ ∈ C ( K ) ∩ M , by the definition of ∼ M , there exists a (necess arily con tin uous) f unction ψ ′ suc h that ψ = ψ ′ ◦ q M . Th us C ( K ) ∩ M ⊆ Y . Let R = { ϕ ∈ C ( K / M ) : ϕ ◦ q M ∈ M } . Then R is a subring of C ( K / M ) whic h separates p o ints and whic h contains all rationa l constants . By the Stone-W eierstrass Theorem, R is dense in C ( K / M ), whic h implies that C ( K ) ∩ M is dense in Y . Observ e that, under t he assumptions of the ab ov e L emma, the norm closure o f C ( K ) ∩ M is p oin t wise closed. Indeed, if f ∈ C ( K ) \ cl( K ∩ M ) then there are x, y ∈ K suc h that x ∼ M y while f ( x ) 6 = f ( y ). Consequen tly , V = { g : g ( x ) 6 = g ( y ) } is a neighborho o d of f in the p oin t wise con v ergence top ology , disjoin t from cl( K ∩ M ). 3.1 Pro jections ind u ced b y elemen tary sub structures Next w e sho w ho w to use elemen tary submo dels for constructing b ounded pro jections. This idea has a lr eady b een applied, in case of WC G spaces, by Koszmider [22]. Lemma 3.3. Assume X is a Ban ach sp ac e, D ⊆ X ∗ is r -n o rming a n d M is an ele- mentary substructur e of a big enough h H ( θ ) , ∈i such that X, D ∈ M . Then 9 (a) X M ∩ ⊥ ( D ∩ M ) = { 0 } ; (b) the c anonic al pr o j e ction P : X M ⊕ ⊥ ( D ∩ M ) → X M has norm 6 r . Pr o of. Fix x ∈ X ∩ M , y ∈ ⊥ ( D ∩ M ) and fix ε > 0. Since D is r -norming, there exists d ∈ D suc h that r | d ( x ) | / k d k > k x k − ε . Since x ∈ M , b y eleme n tarity w e may assume that d ∈ M . Thus d ∈ D ∩ M and d ( y ) = 0. It follo ws that k x k 6 r | d ( x ) | / k d k + ε = r | d ( x + y ) | / k d k + ε 6 r k x + y k + ε. By con tin uit y , w e see t ha t k x k 6 r k x + y k whenev er x ∈ X M and y ∈ ⊥ ( D ∩ M ). In particular, X M ∩ ⊥ ( D ∩ M ) = { 0 } , b ecause if x ∈ X M ∩ ⊥ ( D ∩ M ) then − x ∈ ⊥ ( D ∩ M ) and k x k 6 r k x − x k = 0. Note tha t, in the ab ov e lemma, the subspace X M ⊕ ⊥ ( D ∩ M ) is closed in X . It may happ en that ⊥ ( D ∩ M ) = 0 (consider X = ℓ ∞ ) and in t hat case the a b ov e lemma is meaningless. W e are going t o discuss Banac h spaces for whic h Lemma 3.3 pro vides a w a y to construct full pro jections. 3.2 W CG spaces and Plic hko pairs W e demonstrate the use of elemen tary submo dels f o r finding pro jections in w eakly compactly generated spaces. Recall that a Banac h space is w e ak ly c o mp actly gener ate d (briefly: WCG ) if it con tains a linearly dense w eakly compact set. Prop osition 3.4. L et X b e a we akly c omp actly gener ate d Banac h sp ac e and let θ b e a big enough r e gular c ar dina l. F urther, let M  h H ( θ ) , ∈i b e such that X ∈ M . Then ther e exists a norm one pr oje ction P M : X → X M such that k er( P M ) = ⊥ ( X ∗ ∩ M ) . Pr o of. Let K b e a linearly dense w eakly compact subset of X . By Lemma 3.3, it suffices to ch ec k that X M ∪ ⊥ ( X ∗ ∩ M ) is linearly dense in X . Supp ose ϕ ∈ X ∗ \ { 0 } is suc h that ( X ∩ M ) ⊆ k er( ϕ ) and ⊥ ( X ∗ ∩ M ) ⊆ ke r( ϕ ). The latter inclu sion implies that ϕ ∈ cl ∗ ( X ∗ ∩ M ), b ecause X ∗ ∩ M is Q -linear. Fix p ∈ K suc h that ϕ ( p ) 6 = 0. L et U 0 , U 1 ⊆ R b e disjoint op en ra tional interv als suc h that 0 ∈ U 0 and ϕ ( p ) ∈ U 1 . Let K 0 b e the we ak closure of K ∩ M . Note that ϕ ↾ K 0 = 0, b ecause ϕ is w eakly contin uous. Using the fa ct that ϕ ∈ cl ∗ ( X ∗ ∩ M ) , f o r eac h x ∈ K 0 c ho ose ψ x ∈ X ∗ ∩ M suc h that ψ x ( x ) ∈ U 0 and ψ x ( p ) ∈ U 1 . By compactness, there are x 0 , x 1 , . . . , x n − 1 ∈ K 0 suc h t ha t ( ∗ ) K 0 ⊆ [ i 1 define G n = { s ∈ Γ : k P s k 6 n } . W e claim that one of these sets is cofinal in Γ. Supp ose otherwise and for eac h n ∈ ω c ho ose t n suc h that k P s k > n whenev er t n 6 s . Using the directedness of Γ, construct a sequence s 1 < s 2 < . . . suc h that t n 6 s n for n ∈ ω . Let s ∞ = sup n ∈ ω s n . Then k P s ∞ k = + ∞ , a con tradiction. Fix k > 1 suc h that Γ ′ := G k is cofinal in Γ. W e claim that Γ ′ is also closed. F or fix s 0 < s 1 < . . . in Γ ′ and let t = sup n ∈ ω s n . W e need to show that k P t k 6 k . Supp ose this is not true and fix x ∈ X with k x k = 1 and k P t x k = r > k . Let ε = ( r − k ) / 2. Using the second part of ( 3 ), find m ∈ ω and y ∈ P s m X suc h that k P t x − y k < ε/ k . Note that P s m = P s m ◦ P t . Using the fact that k P s m k 6 k , w e get k P s m x k 6 k and k y − P s m x k = k P s m ( y − P t x ) k 6 k k y − P t x k < ε. Th us k P t x k 6 k P t x − y k + k y − P s m x k + k P s m x k < ε/ k + ε + k 6 2 ε + k = r = k P t x k . This contradiction completes the pro of. By the a b ov e prop osition, we shall alw a ys assume that a pro jectional sk eleton { P s } s ∈ Γ satisfies the condition (4) sup s ∈ Γ k P s k < + ∞ . W e shall sa y that { P s } s ∈ Γ is an r -pr oje ctiona l skeleton if it is a pro jectional sk eleton suc h t ha t k P s k 6 r for ev ery s ∈ Γ. The remaining part of this section will be dev oted to proving basic prop erties of pro jectional ske letons. Lemma 4.2. L et { P s } s ∈ Γ b e a pr oje ctional skeleton i n X and let s 0 < s 1 < . . . b e such that t = sup n ∈ ω s n in Γ . Then P t x = lim n →∞ P s n x for every x ∈ X . 14 Pr o of. Let r = sup s ∈ Γ k P s k and fix x ∈ X , ε > 0 . By the second part of (3 ) , find y ∈ S n ∈ ω P s n X suc h that k P t x − y k < ε/ (1 + r ). Cho ose k suc h that y ∈ P s k X . Note that P t y = y and P s n y = y for n > k . Thus, g iven n > k , w e ha v e k P t x − P s n x k 6 k P t x − y k + k y − P s n x k < ε/ (1 + r ) + k P s n ( y − P t x ) k 6 ε / (1 + r ) + r k y − P t x k < ε/ (1 + r ) + r ε/ (1 + r ) = ε. This shows that lim n →∞ k P t x − P s n x k = 0. Lemma 4.3. L et { P s } s ∈ Γ b e a pr oje ctional skeleton in X and let T ⊆ Γ b e a dir e cte d subset of Γ . Then the formula P T x = lim s ∈ T P s x wel l defines a b ounde d pr oje ction of X onto cl( S s ∈ T P s X ) . Pr o of. W e must sho w that { P s x } s ∈ T is a Cauc h y net f or ev ery x ∈ X . Fix a big enough regular cardinal θ so that all r elev an t ob jects are contained in H ( θ ). Fix a countable elemen tary substructure M of H ( θ ) such that { P s } s ∈ Γ ∈ M a nd T ∈ M . Let S = T ∩ M . By elemen tarity , S is a directed set. Supp ose that there exists x ∈ X suc h tha t { P s x } s ∈ T is no t a Cauc h y net. By elemen tarit y , there exist x ∈ M ∩ X and ε > 0 in M suc h that M | = ∀ t ∈ T ∃ s, s ′ ∈ T ( s > t ∧ s ′ > t ∧ k P s x − P s ′ x k > ε ) . This means in particular that, g iv en t ∈ S , there exist s, s ′ ∈ S suc h that s, s ′ > t and k P s x − P s ′ x k > ε . In order to get a contradiction, it suffices to show that { P s x } s ∈ S is a Cauc h y net. Since S is coun table and directed, it has an increasing cofinal sequence s 0 < s 1 < . . . . Let u = sup n ∈ ω s n . Then u = sup S . By Lemma 4.2, P t x = lim n →∞ P s n x for ev ery x ∈ X . Fix ε > 0 and find k suc h that k P s n x − P s k x k < ε/ (2 r ) for ev ery n > k , where r > 1 is suc h that k P s k 6 r f o r ev ery s ∈ T . Fix s, t ∈ S suc h that s, t > s k . Cho o se ℓ > k so that s, t 6 s ℓ . W e hav e k P s x − P s k x k = k P s P s ℓ x − P s P s k x k 6 k P s k · k P s ℓ x − P s k x k < r ε/ (2 r ) = ε/ 2 . Similarly , k P t x − P s k x k < ε/ 2. Thus , finally k P s x − P t x k < ε for eve ry s, t ∈ S with s, t > s k , i.e. { P s x } s ∈ S is a Cauc h y net for eve ry x ∈ X . This show s that P T is w ell defined. It is clear that P T is a b ounded pro jection and that P T x = x iff x ∈ cl( S s ∈ T P s X ). 15 4.2 Pro jectio nal resolutions of the iden tit y Theorem 4.4. Every Bana c h sp ac e with 1 -pr oje ctional skeleton has a pr oje ctional r es- olution of the identity. Pr o of. Let κ = dens X and let s = { P s } s ∈ Γ b e a pro jectional sk eleton in X suc h that k P s k = 1 for ev ery s ∈ Γ. Fix a con tin uous c hain { T α } α<κ of up-directed subsets of Γ satisfying | T α | 6 α + ℵ 0 for each α and suc h that E = [ { P s X : s ∈ T α , α < κ } is dense in X . Contin uity o f the chain means that T δ = S ξ <δ T ξ whenev er δ is a limit ordinal. Let X α = cl( S s ∈ T α P s X ). Then { X α } α<κ is a chain of closed subspaces o f X , the densit y of X α do es not exceed | α | + ℵ 0 and X δ = cl( S ξ <δ X ξ ) for ev ery limit ordinal δ . By Lemma 4.3, form ula P α x = lim s ∈ T α P s x defines a pro jection of X on to X α . Clearly , k P α k = 1, b ecause k P s k = 1 for s ∈ Γ. It is easy to c hec k, a gain using Lemma 4.3, that P α = P α ◦ P β = P β ◦ P α for α < β . Th us, { P α } α<κ is a PRI on X . Corollary 4.5. Given a Banach sp ac e X of density ℵ 1 , the fol lowing pr op erties ar e e quiva lent. (a) X has a 1 -pr oje ctional skeleton. (b) X has a pr oje ctional r esolution of the identity. Pr o of. Implication (a) = ⇒ (b) follow s fro m the ab o v e theorem. In case of density ℵ 1 , ev ery PRI is a 1-pro jectional sk eleton. 4.3 Norming space induc ed b y a pro jectional sk eleton W e shall now lo ok at the dual of a space with a pro jectional ske leton. Let s = { P s } s ∈ Γ b e a pro jectional sk eleton in a Banac h space X . The set D = [ s ∈ Γ P ∗ s X ∗ is clearly a linear subspace o f X . Notice t ha t P ∗ s X ∗ ∩ B X ∗ endo w ed with the w eak ∗ top ology is second countable, b ecause P ∗ s X ∗ is linearly homeomorphic to the dual of P s X . Let r = sup s ∈ Γ k P s k . Giv en x ∈ S X , there is s ∈ Γ suc h that x = P s x ; c hoo se ϕ ∈ X ∗ suc h that ϕ ( x ) = 1 = k ϕ k . Then ( P ∗ s ϕ ) x = ϕ ( P s x ) = ϕ ( x ) = 1 and k P ∗ s ϕ k 6 r . This sho ws that the space D is r -norming. W e shall sa y that D is the dual norming subspace induc e d by s and w e shall denote it b y D ( s ). 16 Let s = { P s } s ∈ Γ and t = { Q t } t ∈ Π b e pro jectional ske letons in the same Bana ch space X . W e say that s and t ar e e quivalent if they induce the s ame norming subspace, i.e. S s ∈ Γ im P ∗ s = S t ∈ Π im Q ∗ t . It turns out that, with help o f elemen tary submo dels, a pro jectional ske leton can b e reco v ered (up to equiv alence) from the norming space. Lemma 4.6. L et s = { P s } s ∈ Γ b e a pr oje ctional skeleton in a Banach sp ac e X and let D ⊆ D ( s ) b e norm i n g for X . F urther, let θ b e a bi g enough r e gular c ar dinal and let M  h H ( θ ) , ∈i b e c ountable and such that s ∈ M . Then the p r oje ction ind uc e d by h X , D , M i e quals P t , wher e t = sup(Γ ∩ M ) . Pr o of. Observ e that [ s ∈ Γ ∩ M im P s ⊆ cl ( X ∩ M ) . This is b ecause, giv en s ∈ Γ ∩ M , b y elemen tarit y there exists a coun table set A ∈ M whic h is dense in im P s . By Prop osition 3.1(c), A ⊆ X ∩ M , therefore im P s ⊆ cl ( X ∩ M ). It fo llo ws that im P t = cl( S s ∈ Γ ∩ M im P s ) ⊆ cl( X ∩ M ). On the ot her hand, giv en x ∈ X ∩ M , by elemen tarit y there is s ∈ Γ ∩ M suc h that x ∈ im P s ; thus x ∈ im P t . Hence im P t = X M . Notice that, again by elemen tarit y , P ∗ t ϕ = ϕ whenev er ϕ ∈ D ∩ M . Th us ker P t ⊆ ⊥ ( D ∩ M ) , b ecause if P t x = 0 and ϕ ∈ D ∩ M then ϕ ( x ) = ( P ∗ t ϕ ) x = ϕ ( P t x ) = 0. It follo ws that k er P t = ⊥ ( D ∩ M ), because X = X M ⊕ k er P t and X M ∩ ⊥ ( D ∩ M ) = { 0 } (Lemma 3.3(a)). Theorem 4.7. L et X b e a Banach sp ac e and let D ⊆ X ∗ b e an r -no rm ing set ( r > 1 ). The fol lowing pr op erties ar e e quivalent. (a) X has an r -pr oje ctional skel e ton s such that D ⊆ D ( s ) . (b) D gener ates pr oje ctions in X . Pr o of. Implication (a) = ⇒ (b) follo ws f rom Lemma 4.6. (b) = ⇒ (a) F ix a big enough regular cardinal θ and let Γ b e the family of all countable elemen tary substructures M o f h H ( θ ) , ∈i suc h that D ∈ M . Endow Γ with inclusion. Clearly , Γ is a σ -directed p oset. Fix M ∈ Γ and let P M b e the pro jection onto cl( X ∩ M ) with ker( P M ) = ⊥ ( D ∩ M ). By definition, P M is defined on the entire space. F urther, k P M k 6 r , b y L emma 3.3. Giv en a sequence M 0 ⊆ M 1 ⊆ . . . in Γ, the union M = S n ∈ ω M n is an elemen tary substructure of H ( θ ) suc h that cl( X ∩ M ) is the closure of S n ∈ ω cl( X ∩ M n ). It fo llo ws that s = { P M } M ∈ Γ is a pro jectional ske leton in X . It is clear that D ⊆ D ( s ), because D ∩ M ⊆ im P ∗ M . Corollary 4.8. L et X b e a Banach sp ac e with a pr oje ctional skeleton. Then ther e exis ts a r enorming of X under which X has an e quivalent 1 -pr oje ctional skeleton. 17 Pr o of. Let D b e the dual norming subspace induced by a fixed pro jectional sk eleton in X . By Lemma 4.6, D generates pro jections in X . Consider a renorming of X after which D b ecomes 1-nor ming (se e Propo sition 2.1). By Theorem 4.7, X has a 1-pro jectional sk eleton. As a n a pplication of Theorem 4 .7, we prov e that the class of Banac h spaces with a 1-pro jectional sk eleton is stable under arbitrary c 0 - and ℓ p -sums. Theorem 4.9. L et { X α } α<κ b e a c ol le c tion of Ban a ch sp ac es and let X = L α<κ X α b e endowe d either with the c 0 -norm or with ℓ p -norm ( 1 6 p < ∞ ). F urther, assume that for e ach α < κ , D α ⊆ X α is 1 -norming and gen e r ates pr oje ctions in X α . Then the set D = { ϕ ∈ X ∗ : ( ∀ α ) ϕ ↾ X α ∈ D α and |{ α : ϕ ↾ X α 6 = 0 }| 6 ℵ 0 } is 1 -norming and gen e r ates pr oje ctions in X . Pr o of. The f a ct that D is 1-norming follow s from the prop erties of the c 0 -sum and the ℓ p -sum. Define suppt( ϕ ) = { α : ϕ ↾ X α 6 = 0 } . Fix a suitable M  h H ( θ ) , ∈i so that D ∈ M . Let S = κ ∩ M . Note that suppt ( ϕ ) ⊆ S whenev er ϕ ∈ D ∩ M . Supp ose X 6 = X M ⊕ ⊥ ( D ∩ M ) and fix ψ ∈ X ∗ satisfying X ∩ M ⊆ k er ψ and ⊥ ( D ∩ M ) ⊆ k er ψ . Then ψ is in the w eak ∗ closure of the linear hull of D ∩ M , therefore suppt( ψ ) ⊆ S . Assuming ψ 6 = 0, there is α ∈ S suc h that ψ α := ψ ↾ X α 6 = 0. Note tha t X α ∩ M ⊆ ke r ψ α . If x ∈ ⊥ ( D α ∩ M ) and ϕ ∈ D ∩ M then ϕ ↾ X α ∈ D α ∩ M so ϕ ( x ) = 0. It follow s that ⊥ ( D ∩ M ) ∩ X α = ⊥ ( D α ∩ M ) . Th us ⊥ ( D α ∩ M ) ⊆ k er ψ . On the other hand, X α = cl( X α ∩ M ) ⊕ ⊥ ( D α ∩ M ) , b ecause D α ∈ M . This is a contradiction. W e finish this section b y exhibiting a top o lo gical prop ert y of norming spaces induced b y a pro jectional ske leton. Theorem 4.10. L et X b e a B anach sp ac e wi th a pr oje ctional skeleton s = { P s } s ∈ Γ and let D ⊆ X ∗ b e the norming sp ac e induc e d by s , e n dowe d with the w eak ∗ top olo gy. The n : (a) The closur e in X ∗ of every c ountable b ounde d subset o f D is metrizable and c on- taine d in D . (b) D is c ountably tight. Pr o of. P art (a) is trivial: ev ery coun table subset of D is con tained in Y s = P ∗ s X ∗ for some s ∈ Γ and ev ery b ounded subset of Y s with the w eak ∗ top ology is second coun table. (b) Let A ⊆ D and p ∈ cl ∗ ( A ) ∩ D b e giv en. Replacing A by A − p , w e may a ssume that p = 0. Fix a big enough regular cardinal θ and a coun table elemen tary substruc ture M of h H ( θ ) , ∈i suc h that X, s , A ∈ M . W e claim that 0 ∈ cl ∗ ( A ∩ M ). 18 Let t = sup(Γ ∩ M ) and let Y t = P ∗ t X ∗ . Fix a w eak ∗ neigh b orho o d U of p . W e ma y assume that U = T i 0 is rational. By Lemma 4.6, P t X = cl( X ∩ M ). By Bana ch’s Op en Mapping Principle, P t − 1 [ X ∩ M ] is dense in X . Hence, without loss of generality , w e may assume that P t x i ∈ M for eac h i < n . Th us W := T i 0 a nd assume { A α } α<δ has already b een defined. Supp ose first that δ = β + 1. W e use Lemma 5 .2 . Let Y = P δ X a nd let D ′ = { y ↾ Y : y ∈ D } ⊆ Y ∗ . Then h Y , D ′ i is a Plic hk o pair and P β ↾ Y is a pro jection whose dual preserv es D ′ . By Lemma 5.2, there is a linearly dense subset B of k er P β ∩ Y witnessing that h ker P β ∩ Y , E i is a Plic hk o pair, where E = D ′ ∩ k er[( P β ↾ Y ) ∗ ]. It follo ws t ha t suppt( y , B ) is countable whenev er y ∈ D ∩ k er P ∗ β . Define A δ = A β ∪ B . Conditions (i), (ii) and (iv) are ob viously satisfied. It remains to c hec k (iii). Fix y ∈ D . By the induction hypothesis, suppt( y , A β ) is coun table. Let z = y − P ∗ β y . Then z ∈ D ∩ k er P β , so suppt ( z , B ) is coun table. Finally , g iv en b ∈ B w e ha v e z ( b ) = y ( b ) − ( P ∗ β y ) b = y ( b ) − y ( P β b ) = y ( b ) , therefore suppt( y , B ) = suppt( z , B ) is coun table. This show s (iii), b ecause suppt ( y , A δ ) = suppt( y , A β ) ∪ suppt ( y , B ). Supp ose now t hat δ is a limit ordinal. Define A δ = S ξ <δ A ξ . Clearly , conditions (i) and ( ii) are satisfied. Condition (iv) is o b vious, so it remains to c hec k (iii). Fix y ∈ D . There is nothing to prov e if δ has a coun table cofinality , b ecause then suppt( y , A δ ) = S ξ <δ suppt( y , A ξ ). Assume cf δ > ℵ 0 . Since A δ ⊆ P δ X , w e see that suppt( y , A δ ) = suppt( P ∗ δ y , A δ ). Thus , we ma y assume that y = P ∗ δ y . Con tin uit y of the pro jectional sequence implies that y = lim ξ <δ P ∗ ξ y , where the limit is with resp ect to the w eak ∗ top ology . W e kno w that P ∗ δ D is contained in a Σ-space, therefore it is w eak ∗ coun tably tigh t. It follows that t here exis ts α < δ suc h that P ∗ ξ y = y for α 6 ξ < δ . In particular, suppt( y , A δ ) = suppt( y , A α ), b ecause if a ∈ A δ \ A α then y ( a ) = ( P ∗ α y ) a = y ( P α a ) = 0, b y (iv). By the induction hypothesis, suppt( y , A δ ) is coun table. This sho ws (iii) . 21 Finally , set A = S α<κ A α . By (i), A is linearly dense in X . It remains to c hec k that suppt( y , A ) is coun table for ev ery y ∈ D . Fix y ∈ D . By the assumption, y ∈ P ∗ α D for some α < κ . Th us P ∗ ξ y = y whenev er α 6 ξ < κ and we conclude lik e in the limit case of the ab o v e construction. This completes t he pro of. The ab ov e theorem should b e compared with the results of S. G ul ′ k o (see [12, 13, 14]), where similar preserv ation w a s prov ed fo r top ological spaces whic h hav e a con tin uous injection in to a Σ-pro duct. Corollary 5.4. L et X b e a Banach sp ac e with a pr oje ctional se quenc e { P α } α<κ such that P α X is we akly Linde l¨ of determine d for e ach α < κ . Then X is a Plichko sp ac e. Corollary 5.5. Given a Bana c h sp ac e X , the f o l lowing pr op erties ar e e quivalent. (a) X ∗ gener ates pr oje ctions in X . (b) X is we a k ly Lindel¨ of determine d. By a result o f Orihuela, Sc hac herma y er and V aldivia [31 ], the a b ov e prop erties are also equiv alen t to “ h B X ∗ , w eak ∗ i is Corson compact”. It is natural to ask when X (as a subspace of X ∗∗ ) generates pro jections in X ∗ . By a result of F abi´ an and Go defroy [9] this is the case when X is Asplund. Sp ecifically , assuming X is an Asplun d space, the authors o f [9] construct a pro j ectional generator h X , Φ i in X ∗ . On the other hand, Orih uela and V aldivia noted in [32, Thm. 3] that the existence o f a pro jectional generator with domain X and with v alues in X ∗ implies that X is Asplund. Recall that a Banac h space X is Asplund if the dual of ev ery separable subspace of X is separable. Assu me X generates pro jections in X ∗ and fix a separable subspace Y of X . Fix a countable M  H ( θ ) suc h that X ∈ M and Y ∩ M is dense in Y and let P : X ∗ → X ∗ b e the pro jection with im P = cl( X ∗ ∩ M ) and k er P = ( X ∩ M ) ⊥ . Then P ∗ y = y for eve ry y ∈ Y , b ecause Y ⊆ cl ∗ ( X ∩ M ). Fix ϕ ∈ Y ∗ and let ψ ∈ X ∗ b e an extension of ϕ . Then ( P ψ ) y = ψ ( P ∗ y ) = ψ ( y ) = ϕ ( y ) f or ev ery y ∈ Y . Th us , ϕ = ( P ψ ) ↾ Y . It follo ws that Y ∗ is separable b ecause so is im P . Summarizing, we ha v e: Prop osition 5.6. Given a Banach sp ac e X , the fol lowing pr op erties ar e e quivalent. (a) X is Asplund. (b) X gener ates p r oje ctions in X ∗ . 22 5.2 A c haracteriza tion of Plic hk o spaces Theorem 5.7. L et X b e a Banach sp ac e and let r > 1 . The fol low i n g pr op erties ar e e quiva lent. (a) X has a c o m mutative r -pr oje ctiona l skeleton. (b) X is an r -Plichko s p ac e. Pr o of. Implication (b) = ⇒ (a) is con tained in Prop osition 5.1. F o r t he con v erse impli- cation, w e use Theorem 4 .4, Lemma 4.3 and induction on the density of X . Supp ose w e ha v e prov ed that (a) = ⇒ (b) for spaces o f densit y < κ and fix a Banac h space X of den- sit y κ with a comm uta tiv e r - pro jectional sk eleton { P s } s ∈ Γ . By (the pr o of of ) Theorem 4.4, there exists an r -pro jectional resolution of the iden tit y s = { P α } α<κ on X suc h that for each α < κ there is a directed set S α ⊆ Γ with P α x = lim s ∈ S α P s x for x ∈ X (to b e formal, w e need to assume t ha t Γ ∩ κ = ∅ ) . Observ e that, by contin uity , P s ◦ P α = P α ◦ P s holds for ev ery s ∈ Γ and α < κ . Let D b e the norming space induced b y s . Fix y ∈ D and fix α < κ . Let s ∈ Γ b e such that y = P ∗ s y . Then P ∗ α y = P ∗ α P ∗ s y = P ∗ s P ∗ α y ∈ D . Hence P ∗ α D ⊆ D . Now use Theorem 5.3. In case where cf κ = ℵ 0 , it may happen t ha t D 6 = S α<κ P ∗ α D , but we ma y replace D b y S α<κ P ∗ α D , still having an r -norming space. By Theorem 5.3, h X , D i is a Plic hk o pair, therefore X is r - Plic hk o. 6 Spaces of con tin uous functions W e discuss a natural class of compact spaces K for whic h C ( K ) has a pro jectional sk eleton. Let R 0 denote t he class of all compacta whic h ha v e a retractional sk eleton. F ollo wing [25], a r etr actional skeleton (briefly: r-skeleton ) in a compact space K is a family of r e- tractions { r s } s ∈ Γ on to metrizable subsets, indexed b y a n up-directed p oset Γ, satisfying the fo llowing conditions: 1. s 6 t = ⇒ r s = r s ◦ r t = r t ◦ r s . 2. F or ev ery x ∈ X , x = lim s ∈ Γ r s ( x ). 3. Give n s 0 < s 1 < . . . in Γ, t = sup n ∈ ω s n exists and r t ( x ) = lim n →∞ r s n ( x ) for ev ery x ∈ K . It has b een pro v ed in [25] that V aldivia compacta are precisely those compact spaces whic h hav e a comm utativ e r-sk eleton. The ordinal ω 2 + 1 is an example of a space 23 in clas s R 0 whic h is not V aldivia compact. It is clear that ev ery r-sk eleton induces a 1-pro jectional sk eleton on the space of con tin uous functions; that is: Prop osition 6.1. L et K b e a c omp act sp ac e wi th a r etr actional skeleton { r s } s ∈ Γ . Then { r ∗ s } s ∈ Γ is a pr oje ctional skeleton in C ( K ) , wher e r ∗ s denotes the tr ansformation adjoint to r s . A s imple application of Lemma 4.3 sho ws that ev ery space from class R 0 can be de- comp osed in to a con tin uous inv erse seque nce of retractions onto smaller spaces in class R 0 (notion dual to a PRI). This show s that R 0 ⊆ R , where R is the smallest class of spaces containing all metric compacta and closed under limits of con tin uous in v erse sequence s of retractions (see [6, 23]). Note that class R 0 restricted to spaces of we ight 6 ℵ 1 coincides with the class of V aldivia compacta ([25, Cor. 4.3]). This is no t the case with class R (see [25, Example 4.6(b)]), therefore R 0 6 = R . Let us a dmit that the con v erse to Prop osition 6.1 fails, namely t here exist compact spaces K / ∈ R suc h that C ( K ) is 1-Plic hk o, see [3]. On the other hand, b y Lemma 4.2, w e ha v e: Prop osition 6.2. L et D ⊆ X ∗ b e a 1 -normin g sp ac e whic h gene r ates pr oj e ctions in a Banach sp ac e X . Then B X ∗ endowe d with the w eak ∗ top olo gy b elongs to cla ss R 0 . The f ollo wing results are dual to Theorems 4.7, 4 .9 and 4.1 0 resp ectiv ely . Theorem 6.3. L et K b e a c omp act sp ac e and let D ⊆ K b e a den s e c ountably close d set. The fol lowing pr op erties ar e e quivalent: (a) K ∈ R 0 and D is induc e d by an r-sk eleton in K . (b) F or ev e ry sufficiently big c ar dinal θ , for every c ountable elem entary substructur e M of H ( θ ) with K, D ∈ M , the quotient q M K : K → K / M r estricte d to cl( D ∩ M ) is one-to-one. Pr o of. Assume (a) and fix a coun table M  h H ( θ ) , ∈i suc h that K, D ∈ M . By elemen- tarit y , there exists { r s } s ∈ Γ ∈ M whic h is a n r- sk eleton in K suc h that D = S s ∈ Γ r s [ K ]. Fix x, y ∈ cl( D ∩ M ), x 6 = y . Let t = sup(Γ ∩ M ). By elemen tarity , D ∩ M ⊆ r t [ K ], therefore also cl( D ∩ M ) ⊆ r t [ K ]. It fo llo ws that x = r t ( x ) and y = r t ( y ). Let { s n } n ∈ ω ⊆ Γ ∩ M b e increasing and suc h that t = sup n ∈ ω s n . Then x = lim n →∞ r s n ( x ) and y = lim n →∞ r s n ( y ). It follo ws that r s k ( x ) 6 = r s k ( y ) for all but finitely man y k ∈ ω . Since r s k ∈ M and r s k [ K ] is second coun table, w e deduce that x 6∼ M y . This sho ws that (a) = ⇒ (b). The pro of o f (b) = ⇒ (a) is similar to that of (b) = ⇒ ( a ) in Theorem 4.7: the family Γ of all coun table M  h H ( θ ) , ∈i with K , D ∈ M is an r-ske leton on K , since each q M K can b e treated as a r etra ction o f K o n to cl( K ∩ M ). 24 Note that coun tably tigh t spaces in class R 0 are precisely Corson compacta. Th us, in case K is Corson compact, we hav e D = K and the ab ov e theorem giv es Bandlo w’s c haracterization [4]. Prop osition 6.4. Class R 0 is close d under arbitr ary pr o ducts. Pr o of. Let { K α : α ∈ κ } b e a family of spaces in R 0 and let K = Q α ∈ κ K α . F o r each α ∈ κ c ho ose a dense set D α ⊆ K α whic h is induced by a fixed r-sk eleton in K α . Fix p ∈ Q α ∈ κ D α and let D b e the Σ-pro duct of { D α } α ∈ κ based on p , i.e. D = { x ∈ Y α ∈ κ D α : | suppt p ( x ) | 6 ℵ 0 } , where suppt p ( x ) := { α ∈ κ : x ( α ) 6 = p ( α ) } . It is clear tha t D is countably closed and dense in K . W e che c k condition (b) of Theorem 6.3 . L et θ > κ b e a regular cardinal suc h that fo r ev ery α < κ statemen t (b) of Theorem 6.3 holds for eve ry coun table M  H ( θ ) with K α , D α ∈ M . Fix a countable M  H ( θ ) with { K α } α ∈ κ , { D α } α ∈ κ ∈ M . Let S = κ ∩ M . Observ e that cl( D ∩ M ) ⊆ { x ∈ K : suppt p ( x ) ⊆ S } . Fix x 6 = y in cl( D ∩ M ). Then x ( α ) 6 = y ( α ) for some α . By the ab ov e remark, α ∈ M . Th us K α , D α ∈ M and therefore q M K α is one-to-one on cl( D α ∩ M ). Finally , if f ∈ C ( K α ) ∩ M separates x ( α ) fr om y ( α ), then g = f ◦ pr α ∈ M and g ( x ) 6 = g ( y ), where pr α denotes the pro jection on to the α -th co ordinate. This sho ws that q M K is one-to -one on cl( D ∩ M ). By Theorem 6.3, K ∈ R 0 . Theorem 6.5. Assume { R s } s ∈ Γ is an r-skeleton in a c omp a c t sp ac e K and let D = S s ∈ Γ R s [ K ] . Then (1) D is dense in K and for every c ountable set A ⊆ D the closur e cl K ( A ) i s metriz- able and c on taine d in D . (2) D has a c ountable tightness. (3) D is a normal sp ac e and K = β D . W e shall sa y that D is the dense set ind uc e d by { R s } s ∈ Γ . Pr o of. (1) follo ws fro m the σ - directedness o f Γ: ev ery coun table subset of D is con tained in some K s := R s [ K ]. The countable tightne ss of D follo ws fr om Theorem 4.10(b), b ecause D ⊆ C ( K ) ∗ generates pro jections in C ( K ). 25 It remains to prov e that D is a normal space and that K = β D . Fix disjoin t relativ ely closed sets A, B ⊆ D . W e claim that cl K ( A ) ∩ cl K ( B ) = ∅ . This will also sho w that K = β D . Supp ose p ∈ cl K ( A ) ∩ cl K ( B ) and fix a countable M  H ( θ ) (where θ is sufficien tly big) such that A, B , p, { R s } s ∈ Γ ∈ M . Let δ = sup(Γ ∩ M ). Then R δ ( p ) ∈ D , so w e ma y assume tha t R δ ( p ) / ∈ A (inte rc hanging the roles of A and B , if necessary). Recall that K δ := R δ [ K ] is the limit of in v erse system h K s , R t s , Γ ∩ M i , where R t s = R s ↾ K t (see [25, Lemma 3.4]). Th us, there are t ∈ Γ ∩ M and an op en set V ⊆ K t suc h that U := K δ ∩ ( R t ) − 1 [ V ] is a neigh b orho o d of p in K δ disjoin t from A . On the other hand, ( R t ) − 1 [ V ] ∩ A 6 = ∅ . Since K t is second coun table, w e ma y assume that V ∈ M . Th us, by elemen tarit y , there is a ∈ M suc h tha t a ∈ ( R t ) − 1 [ V ] ∩ A . Finally , a ∈ K δ , so a ∈ U ∩ A , a con tradiction. A Bana c h space analogue of part (3 ) of the ab o v e result lo oks as follo ws. Prop osition 6 .6. L et D b e a norm ing sp ac e induc e d by a pr o j e ctional skeleton { P s } s ∈ Γ in a Banach sp ac e X . Then fo r every w eak ∗ c on tinuous function f : D → R ther e exists t ∈ Γ such that f = f ◦ P ∗ t ↾ D . Pr o of. Fix n > 0 a nd consider K n = n B X ∗ . Then { P ∗ s ↾ K n } s ∈ Γ is a retractional sk eleton in K n . By [25, Lemma 5.1], there exists s n ∈ Γ such that f n = f ◦ P ∗ s n ↾ ( D ∩ K n ). W e ma y assume that s 1 6 s 2 6 . . . . Let t = sup n ∈ ω s n . Then f = f ◦ P ∗ t ↾ D . W e are now able to determine w eak ∗ compact subsets of spaces induced b y pro jectional sk eletons. Theorem 6.7. Assume D ⊆ X ∗ gener ates pr oje ctions in a Banach sp ac e X . Then every c omp act subset of D is Corson. Pr o of. Let K ⊆ D b e compact with respect to the w eak ∗ top ology . W e use Bandlo w’s c haracterization [4], whic h is a sp ecial case of Theorem 6.3. Fix a suitable coun table M  h H ( θ ) , ∈i and fix p 6 = q in cl ∗ ( K ∩ M ). Then there is x ∈ X suc h that p ( x ) 6 = q ( x ). Since h X, D i has Prop ert y Ω (see Prop osition 3.7), there exists y ∈ cl( X ∩ M ) suc h that x − y ∈ ⊥ ( D ∩ M ) . In particular, p ( x ) = p ( y ) and q ( x ) = q ( y ). Now , the contin uit y of p and q , find z ∈ X ∩ M suc h tha t p ( z ) 6 = q ( z ). Th us the function ϕ 7→ ϕ ( z ) is a n elemen t of M whic h separates p and q . This shows that p 6∼ M q . Finally , Bandlo w’s theorem [4] (or a special case of Theorem 6.3) sho ws that K is Corson. W e do not know whether the conv erse holds, namely whether ev ery norming w eak ∗ Corson compact set g enerates pro jections, see Question 5. A preserv atio n theorem for V aldivia compacta, dua l to Theorem 5.3, lo oks a s follo ws. W e omit its pro of, since it can b e easily deduced fro m (the pro of of ) Theorem 5.3. 26 Giv en a V aldivia compact K , let us call h K , D i a V aldivi a p air if D is dense in K and there is an embedding j : K → [0 , 1] κ suc h t ha t j [ D ] ⊆ Σ( κ ). Theorem 6.8. L et { r α } α<κ b e a c ontinuous r etr active se q uenc e in a c omp act sp ac e K . L et D ⊆ K b e a dense s et s uch that for e ach α < κ , h r α [ K ] , r α [ D ] i is a V aldivia p air and r α [ D ] ⊆ D for every α < κ . Th en h K , D i is a V aldivia p air. The ab ov e resu lt leads to anot her pro of of [25 , Thm. 6.1], sa ying that a compact space with a comm utativ e r-sk eleton is V aldivia compact. Another corollary is the follo wing. Corollary 6.9. The limit of a c ontinuous r etr active inverse se quenc e of C o rson c om- p a c ta is V aldivi a c omp act. 7 Final remarks and op en p r o blems As w e ha v e already men tioned, the ordinal ω 2 + 1 prov ides an example of a compact space in class R 0 whose space of con tin uous functions is not Plic hk o. An r-sk eleton in ω 2 + 1 can be constructed as follo ws. Denote by Γ the family of all coun table closed subsets A of ω 2 suc h that 0 ∈ A and ev ery isolated p oint of A is isolated in ω 2 . Giv en A ∈ Γ, define r A : ω 2 + 1 → ω 2 + 1 b y setting r A ( α ) = ma x ( A ∩ [0 , α ]). It is straight to c hec k that r A is a retraction on to A (contin uity fo llo ws from the assumption concerning isolated p oin ts). It is easy to che c k that t = { r A } A ∈ Γ is a n r-sk eleton. Obviously , this sk eleton is not comm utativ e. On the other hand, the dual C ( ω 2 + 1) ∗ is 1-Plic hk o (see [19, Example 4.10(a)] or [18, Example 6.10]). Example 7.1. There exists a 1- pro jectional sk eleton s on ℓ 1 ( ω 2 ) suc h that D ( s ) is not a Σ-space, i.e. h ℓ 1 ( ω 2 ) , D ( s ) i is not a Plic hk o pair. Pr o of. W e shall use ω 2 + 1 instead of ω 2 as the co o rdinate set, b ecause ω 2 + 1 = [0 , ω 2 ] has the maximal elemen t with resp ect to the natural w ell o rder. Let Γ consist of all coun table subsets S of ω 2 + 1 such that ω 2 ∈ S . Giv en S ∈ Γ, define f S : ω 2 + 1 → ω 2 + 1 b y f S ( α ) = min( S ∩ [ α, ω 2 ]). Note that f S is generally discon t inuous with respect to the in terv a l top ology on ω 2 + 1. F urther, define Q S : ℓ 1 ( ω 2 + 1) → ℓ 1 ( ω 2 + 1) by setting ( Q S x )( α ) = P ξ ∈ f − 1 S ( α ) x ( ξ ). It is clear that Q S is a we ll defined linear pro jection on to ℓ 1 ( S ) ⊆ ℓ 1 ( ω 2 + 1). F urther, k Q S k = 1 . Fix S ⊆ T in Γ. Clearly , Q T ◦ Q S = Q S , b ecause Q T is iden tit y on ℓ 1 ( T ) ⊇ ℓ 1 ( S ). No w observ e that f S ◦ f T = f S . Th us, giv en x ∈ ℓ 1 ( ω 2 + 1) and α ∈ ω 2 + 1, w e ha v e that f − 1 S ( α ) = S ξ ∈ f − 1 S ( α ) f − 1 T ( ξ ) and conseque n tly ( Q S Q T x )( α ) = X ξ ∈ f − 1 S ( α ) ( Q T x )( ξ ) = X ξ ∈ f − 1 S ( α ) X η ∈ f − 1 T ( ξ ) x ( η ) = X ξ ∈ f − 1 S ( α ) x ( ξ ) = ( Q S x )( α ) . 27 Finally , give n S 0 ⊆ S 1 ⊆ . . . in Γ , the set S ∞ = S n ∈ ω S n is an elemen t of Γ and S n ∈ ω im( Q S n ) is clearly dense in im( Q S ∞ ). It follows that s = { Q S } S ∈ Γ is a pro jectional sk eleton in ℓ 1 ( ω 2 + 1). No w supp ose that h ℓ 1 ( ω 2 ) , D ( s ) i is a Plic hk o pair. By Theorem 5.7, there exists a comm utativ e pro j ectional sk ele ton t = { P t } t ∈ ∆ suc h that D ( t ) = D ( s ). By Lemma 4.6, there exists a cofinal subs et Γ ′ ⊆ Γ suc h that for ev ery S ∈ Γ ′ there is t = t ( S ) ∈ ∆ with Q S = P t . In particular, Q S ◦ Q T = Q T ◦ Q S whenev er S, T ∈ Γ ′ . W e shall deriv e a con tradiction b y finding S, T ∈ Γ ′ suc h t ha t Q S ◦ Q T 6 = Q T ◦ Q S . Giv en S ∈ Γ ′ , define ϕ ( S ) = sup( S ∩ ω 2 ). Construct a c hain { S α } α<ω 1 in Γ ′ so that ϕ ( α ) < ϕ ( β ) whenev er α < β . This is p ossible , b ecause Γ ′ is cofinal in Γ. L et δ = sup α<ω 1 ϕ ( α ). F ix T ∈ Γ ′ suc h that ϕ ( T ) > δ . Then sup( T ∩ δ ) < δ , b ecause δ has cofinalit y ω 1 . Find α < ω 1 suc h that sup( T ∩ δ ) < ϕ ( S α ) < δ . Let S = S α . Fix ξ ∈ S α suc h that sup( T ∩ δ ) < ξ . Then f T ( ξ ) > δ and hence f S f T ( ξ ) = ω 2 , b ecause ϕ ( S ) < δ . On the other hand, f S ( ξ ) = ξ < ϕ ( T ) and hence f T f S ( ξ ) 6 ϕ ( T ) < ω 2 . It follo ws that f S f T ( ξ ) 6 = f T f S ( ξ ). Considering the c haracteristic function of { ξ } as an elemen t o f ℓ 1 ( ω 2 + 1), w e conclude that Q S ◦ Q T 6 = Q T ◦ Q S . Giv en a norming space D ⊆ X ∗ , let T D denote the top ology on X induced b y D , i.e. T D = σ ( X, D ). It can b e sho wn that h X , T D i is Lindel¨ of, whenev er D generates pro jections in X . On the other hand, b y the result of Kalenda [16], D is a Σ-space iff h X , T D i is primarily L indel¨ o f and D ∩ B X ∗ is w eak ∗ coun tably compact. Pr oble m 1 . F ind a top o lo gical prop erty of h X , T D i whic h sa ys when D generates pro- jections in X . Question 1 . Assume X has a 1-pro jectional sk eleton. Do es X hav e a pro jectional gen- erator? Question 2 . Assume X is a Bana c h space of densit y > ℵ 1 and F is a directed family of 1- complemen ted separable subspaces suc h that X = S F and cl( S n ∈ ω F n ) ∈ F whenev er { F n : n ∈ ω } ⊆ F . Do es X necessarily ha v e a pro jectional sk eleton? Note that if X has densit y ℵ 1 then the ab ov e assumptions imply the existence of a PRI, see [24 , Lemma 6.1]. Question 3 . Let X b e a Banach space with a pro jectional sk ele ton. D o es eve ry closed subspace of X ha v e the separable compleme n tation prop erty? Note that, b y the main result of [24], a closed subspace o f a Plic hk o space o f densit y ℵ 1 ma y not ha v e a pro jectional sk eleton. The f ollo wing ques tion has a lready b een ask ed by Ond ˇ rej Kalenda [20]. Question 4 . Is C ( ω 2 + 1) em beddable into a Plichk o space? 28 Question 5 . Assume K ⊆ X ∗ is Corson compact and norming for X . Do es K generate pro jections in X ? If K is a Corson compact in the dua l of a Banach space X and K is norming for X , then X em b eds in to C ( K ). Th us, if C ( K ) is WLD then so is X and conse quen tly K generates pro jections in X . It follows t ha t the abov e question at least consisten tly has affirmativ e answ er: it is relativ ely consisten t with the usual axioms of set theory that C ( K ) is WLD for ev ery Corson compact K (se e e.g. [2, Remark 3.2.3)]). 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