Characterization of (semi-)Eberlein compacta using retractional skeletons
aa r X i v : . [ m a t h . GN ] S e p CHARACTERIZATION OF (SEMI-)EBERLEIN COMPACTAUSING RETRACTIONAL SKELETONS
CLAUDIA CORREA, MAREK C ´UTH, AND JACOPO SOMAGLIA
Abstract.
We deeply study retractions associated to suitable modelsin compact spaces admitting a retractional skeleton and find several in-teresting consequences. Most importantly, we provide a new character-ization of Valdivia compacta using the notion of retractional skeletons,which seems to be helpful when characterizing its subclasses. This char-acterization of Valdivia compacta allowed us to characterize Eberleinand semi-Eberlein compacta in terms of retractional skeletons. Introduction
The study of the class of compact spaces that admit a retractional skeletonwas initiated in [23], where the authors proved that a compact space is Val-divia if and only if it admits a commutative retractional skeleton. Later, in[21] a notion similar to retractional skeletons in the context of Banach spaceswas introduced; namely, the notion of projectional skeletons. In some sense,those notions are dual to each other. More precisely, if a compact space K admits a retractional skeleton, then C ( K ) admits a projectional skeletonand if a Banach space X admits a projectional skeleton, then ( B X ∗ , w ∗ )admits a retractional skeleton. The class of Banach (compact) spaces witha projectional (retractional) skeleton was deeply investigated from variousperspectives and nowadays we have quite a rich family of natural examplesand interesting results related to various fields of mathematics such as topol-ogy [25], Banach space theory [14], theory of von Neumann algebras [3] or J BW ∗ -triples [4]. Let us note that, quite surprisingly, there was indepen-dently introduced also the notion of monotonically retractable topologicalspaces which turned out to be very closely related to the study of compactspaces that admit a retractional skeleton, see [10], and from there on, several Mathematics Subject Classification.
Key words and phrases. retractional skeleton, projectional skeleton, Eberlein compact,semi-Eberlein compact.C. Correa has been partially supported by Funda¸c˜ao de Amparo `a Pesquisa do Es-tado de S˜ao Paulo (FAPESP) grants 2018/09797-2 and 2019/08515-6. M. C´uth has beensupported by Charles University Research program No. UNCE/SCI/023. J. Somagliahas been supported by Universit`a degli Studi di Milano, Research Support Plan 2019and by Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Applicazioni(GNAMPA) of Istituto Nazionale di Alta Matematica (INdAM), Italy. results and modifications of the corresponding notions were considered, seee.g. [5, 16, 17].One of the recent streams in the area is to describe some classes of Banach(compact) spaces using the notion of projectional (retractional) skeletons,see e.g. [23, 8, 14, 21] where the characterizations of Plichko spaces (andValdivia compacta), WLD spaces (and Corson compacta), Asplund spaces,WLD+Asplund spaces and WCG spaces were given.The main two results of this paper (Theorems A and B) are characteriza-tions of Eberlein and semi-Eberlein compacta, respectively, using the notionof retractional skeletons. Let us recall that given a set I we define c ( I ) := { x ∈ R I : ( ∀ ε > |{ i ∈ I : | x ( i ) | > ε }| < ω } ⊂ R I and that a compact space K is Eberlein if it homeomophically embeds into c ( I ), for some set I . This is a central concept in Banach space theory, as itis known that a compact space is Eberlein if and only if it is homeomorphicto a weakly compact set of a Banach space, see [1] or [13, Corollary 13.19].For the notion of shrinkingness we refer the reader to Definition 29. Theorem A.
Let K be a compact space. Then the following conditions areequivalent:(1) K is Eberlein.(2) There exist a bounded set A ⊂ C ( K ) separating the points of K anda retractional skeleton s = ( r s ) s ∈ Γ on K such that(a) s is A -shrinking,(b) f ◦ r s ∈ A , for every f ∈ A and s ∈ Γ . Recall that a compact space K is Eberlein if and only if C ( K ) is WCG,thus Theorem A is naturally connected to the characterization of WCG Ba-nach spaces presented in [14]. The first advantage of our approach is thatTheorem A may be thought of as a topological counterpart and in a certainsense also strengthening of [14, Theorem 21] in the context of C ( K ) spaces,see Remark 33. The second advantage is that quite many steps of our proofseem to be much more flexible and we believe that those may be used inorder to find characterizations of other natural subclasses of Valdivia com-pacta (the most important in this respect is probably Theorem C mentionedbelow). This is witnessed by the characterization of semi-Eberlein compactapresented in Theorem B. Recall that, following [22], we say a compact space K is semi-Eberlein if there exists a homeomorphic embedding h : K → R I such that h − [ c ( I )] is dense in K . We denote by D ( s ) the set induced bya retractional skeleton s (see Definition 1). Theorem B.
Let K be a compact space. Then the following conditions areequivalent:(1) K is semi-Eberlein.(2) There exist a dense subset D ⊂ K , a bounded set A ⊂ C ( K ) sepa-rating the points of K and a retractional skeleton s = ( r s ) s ∈ Γ on K with D ⊂ D ( s ) such that HARACTERIZATION OF EBERLEIN COMPACTA 3 (a) s is A -shrinking with respect to D (b) f ◦ r s ∈ A , for every f ∈ A and s ∈ Γ .(c) lim s ∈ Γ ′ r s ( x ) ∈ D , for every x ∈ D and every up-directed subset Γ ′ of Γ . As mentioned previously, many steps of the proofs of Theorem A andTheorem B are of independent interest and we believe those could be usedwhen trying to characterize other subclasses of Valdivia compacta, whichopens quite a wide area of potential further research. This is outlined inSection 6.Let us now briefly describe the content of each section, emphasizing thegeneral steps mentioned above.Section 2 contains basic notations and some preliminary results.In Section 3 we consider retractions associated to (not necessary count-able) suitable models. The most important outcome is Theorem 15, wherewe summarize the properties of canonical retractions associated to suitablemodels. As an easy consequence, in Proposition 17 we show a very generalmethod of obtaining a continuous chain of retractions on a compact spaceadmitting a retractional skeleton. This part is essentially known as similarresults were obtained e.g. in [5, Lemma 2.5] (using other methods thansuitable models), but our approach is in a certain sense much more flexi-ble (most importantly, because it may be combined with other statementsinvolving suitable models) and we actually use this flexibility later. As acorollary of our investigations we show in Theorem 21 that we may in acertain way combine properties of countably many retractional skeletons.In Section 4, inspired by the proof of [5, Theorem 2.6], we aim at seeing asconcretely as possible the “Valdivia embedding” of compact spaces with acommutative retractional skeleton. As a consequence we obtain the followingresult which might be thought of as the third main result of the whole paper.The most important part which we use later is the implication (i) ⇒ (iv). Theorem C.
Let K be a compact space and s = ( r s ) s ∈ Γ be a retractionalskeleton on K . Then the following conditions are equivalent.(i) D ( s ) is induced by a commutative retractional skeleton.(ii) There exists a subskeleton of s which is commutative.(iii) There exist a subskeleton s = ( r s ) s ∈ Γ ′ of s and a dense set D ⊂ D ( s ) such that for every up-directed set Γ ′′ ⊂ Γ ′ and every x ∈ D we have lim s ∈ Γ ′′ r s ( x ) ∈ D .Moreover, if λ ≥ and A ⊂ λB C ( K ) is a closed, symmetric and convex setseparating the points of K such that f ◦ r s ∈ A , for every f ∈ A and s ∈ Γ ,then those conditions are also equivalent to the following one.(iv) There exists H ⊂ A such that the mapping ϕ : K → [ − , H definedas ϕ ( x )( h ) := hλ ( x ) , for every h ∈ H and x ∈ K , is a homeomorphicembedding and ϕ [ D ( s )] ⊂ Σ( H ) . C. CORREA, M. C ´UTH, AND J. SOMAGLIA
Note that Theorem C provides a characterization of Valdivia compacta,since a compact space is Valdivia if and only if it admits a commutativeretractional skeleton. The proof of Theorem C (i) ⇒ (iv) is done by induction.In the case when the weight of the compact space is ω we have even a muchbetter result which is very constructive, see Theorem 25.In Section 5 we prove (slightly more general versions of) Theorem A andTheorem B. In the case when the weight of the compact space is ω we haveeven a better result, see Corollary 32.Section 6 is devoted to open problems and remarks.2. Notation and preliminary results
We use standard notations from topology and Banach space theory as canbe found in [11] and [13].For a set I , we defineΣ( I ) := { x ∈ R I : | suppt( x ) | ≤ ω } , where suppt( x ) = { i ∈ I : x ( i ) = 0 } is the support of x . Given a subset S of I we denote the characteristic function of S by 1 S .All topological spaces are assumed to be Tychonoff. Let T be a topologicalspace. A subset S ⊂ T is said to be countably closed if C ⊂ S , for everycountable subset C ⊂ S . We denote by w( T ) the weight of T , by C ( T, T )the set of continuous functions from T to T and by βT the ˇCech-Stonecompactification of T . If T is compact, then as usual C ( T ) denotes theBanach algebra of real-valued continuous functions defined on T , endowedwith the supremum norm. Moreover, if A ⊂ C ( T ), we denote by alg( A ) thealgebraic hull of A in the algebra C ( T ). Recall that a compact space T issaid to be Valdivia if there is a homeomorphic embedding h : T → R I suchthat h − [Σ( I )] is dense in T , we refer to [19] for a survey in this subject.Let (Γ , ≤ ) be an up-directed partially ordered set. We say that a sequence( s n ) n ∈ ω of elements of Γ is increasing if s n ≤ s n +1 , for every n ∈ ω . Wesay that Γ is σ -complete if for every increasing sequence ( s n ) n ∈ ω in Γ thereexists sup n s n in Γ. We say that Γ ′ ⊂ Γ is cofinal in Γ if for every s ∈ Γthere is s ∈ Γ ′ with s ≥ s . If Γ is σ -complete and A ⊂ Γ, we denote by A σ the smallest σ -complete subset of Γ containing A . Notice that, by [20,Proposition 2.3], if A is up-directed, then A σ is up-directed. Definition 1.
Following [10], a retractional skeleton in a countably compactspace K is a family of continuous retractions s = ( r s ) s ∈ Γ on K indexed byan up-directed, σ -complete partially ordered set Γ, such that:(i) r s [ K ] is a metrizable compact space for each s ∈ Γ,(ii) s, t ∈ Γ, s ≤ t then r s = r t ◦ r s = r s ◦ r t ,(iii) given an increasing sequence ( s n ) n ∈ ω in Γ, if s = sup n ∈ ω s n ∈ Γ,then r s ( x ) = lim n →∞ r s n ( x ), for every x ∈ K ,(iv) for every x ∈ K , x = lim s ∈ Γ r s ( x ). HARACTERIZATION OF EBERLEIN COMPACTA 5
We say that S s ∈ Γ r s [ K ] is the set induced by the retractional skeleton s andwe denote it by D ( s ). We say that s is commutative if we have r s ◦ r t = r t ◦ r s for every s, t ∈ Γ. We say that s is full if D ( s ) = K .The following preliminary result will be used in what follows quite fre-quently. It seems to be new even though it could be known to some expertsas well. Lemma 2.
Let K be a compact space. Suppose that K has a retractionalskeleton s = ( r s ) s ∈ Γ . Let Γ ′ ⊂ Γ be an up-directed subset, then the mapping R Γ ′ : K → K defined by R Γ ′ ( x ) = lim s ∈ Γ ′ r s ( x ) is a continuous retraction.Moreover, the following holds.(i) If Γ ′ is countable, then s = sup Γ ′ exists and we have R Γ ′ = r s .(ii) If M is an up-directed subset of P (Γ) such that each M ∈ M isup-directed. Then lim M ∈M R M ( x ) = R S M ( x ) , x ∈ K .(iii) For every s ∈ (Γ ′ ) σ we have that r s [ K ] ⊂ R Γ ′ [ K ] .(iv) ( r s | R Γ ′ [ K ] ) s ∈ (Γ ′ ) σ is a retractional skeleton on R Γ ′ [ K ] with inducedset D ( s ) ∩ R Γ ′ [ K ] .(v) If s is commutative, then D ( s ) ∩ R Γ ′ [ K ] = R Γ ′ [ D ( s )] .Proof. Let us start by proving that the mapping R Γ ′ is well-defined. Inorder to do that fix x ∈ K and suppose that ( r s ( x )) s ∈ Γ ′ is an infinite set(otherwise the assertion would be trivial). Since K is compact, there existsa cluster point x ∈ K for the net ( r s ( x )) s ∈ Γ ′ . Let us show that such acluster point x is unique. Indeed, let x = x be two cluster points of( r s ( x )) s ∈ Γ ′ . Let U , U ⊂ K be two open subsets such that x ∈ U , x ∈ U and U ∩ U = ∅ . Let ( s n ) n<ω , ( t n ) n<ω ⊂ Γ ′ be two increasing sequences ofindexes such that s n ≤ t n ≤ s n +1 , r s n ( x ) ∈ U , and r t n ( x ) ∈ U , for every n ∈ ω . Since Γ is σ -complete, we have that sup n ∈ ω s n = sup n ∈ ω t n = s ∈ Γ.Then r s ( x ) ∈ U ∩ U , a contradiction. Therefore R Γ ′ is well-defined.The map R Γ ′ is continuous. Indeed, let ( x λ ) λ ∈ Λ be a net converging to x ∈ K . Up to taking a subnet we may assume without loss of general-ity that R Γ ′ ( x λ ) converges to y . Suppose by contradiction y = R Γ ′ ( x ),then there are two open subsets U, V ⊂ K with y ∈ U and R Γ ′ ( x ) ∈ V ,such that U ∩ V = ∅ . We find recursively two increasing sequences of in-dexes ( s n ) n<ω in Γ ′ and ( λ n ) n<ω in Λ such that r s k ( x λ i ) ∈ V if i ≥ k and r s k ( x λ i ) ∈ U if i < k .Let us sketch the recursion here. Since R Γ ′ ( x λ ) → y , there exists λ ∈ Λsuch that R Γ ′ ( x λ ) ∈ U for every λ ≥ λ . Since r s ( x ) Γ ′ → R Γ ′ ( x ) , there exists s ∈ Γ ′ such that r t ( x ) ∈ V for every t ≥ s . Since r s ( x λ ) Γ ′ → R Γ ′ ( x λ ) ∈ U ,there exists s ≥ s such that r t ( x λ ) ∈ U for every t ≥ s . By the continuityof r s , we have r s ( x λ ) → r s ( x ) ∈ V ; hence there exists λ ≥ λ such that r s ( x λ ) ∈ V for every λ ≥ λ . We proceed recursively in an obvious way.Since Γ is σ -complete, s = sup k ∈ ω s k belongs to Γ. Hence r s k ( x λ i ) con-verges to r s ( x λ i ) ∈ U for every i ∈ ω . Moreover, by compactness we C. CORREA, M. C ´UTH, AND J. SOMAGLIA have T k ∈ ω ( x λ i ) i ≥ k = ∅ , so we may pick ˜ x ∈ T k ∈ ω ( x λ i ) i ≥ k . We observethat r s k (˜ x ) ∈ V for every k ∈ ω , hence r s (˜ x ) ∈ V . On the other hand r s k ( x λ i ) → r s ( x λ i ) ∈ U for every i ∈ ω ; therefore r s (˜ x ) ∈ U , a contradiction.Thus, R Γ ′ is continuous.Finally, R Γ ′ is a retraction. Indeed, pick x ∈ K . Then R Γ ′ ( R Γ ′ ( x )) = lim t ∈ Γ ′ r t ( lim s ∈ Γ ′ r s ( x )) = lim t ∈ Γ ′ lim s ∈ Γ ′ r t ( r s ( x ))= lim t ∈ Γ ′ lim s ∈ Γ ′ ,s ≥ t r t ( r s ( x )) = lim t ∈ Γ ′ r t ( x ) = R Γ ′ ( x ) . It remains to prove the “Moreover” part.(i): If Γ ′ is countable, then we can find an increasing sequence ( s n ) n ∈ ω fromΓ with sup n s n = s = sup Γ ′ . Then, using that the sequence { s n : n ∈ ω } iscofinal in Γ ′ , we obtain R Γ ′ = R { s n : n ∈ ω } = r s .(ii): Suppose that M ⊂ P (Γ) is up-directed and that each M ∈ M isup-directed. Put M ∞ := S M ∈M M , fix x ∈ K and open set U such that R M ∞ ( x ) ∈ U . Let V be an open neighborhood of R M ∞ ( x ) such that V ⊂ U .Then, there exists s ∈ M ∞ such that r s ( x ) ∈ V , for every s ∈ M ∞ with s ≥ s . By the definition of M ∞ , there exists M ∈ M such that s ∈ M . If M ∈ M and M ⊂ M , then s ∈ M . This implies that the set { s ∈ M : s ≥ s } is cofinal in M and so we have R M ( x ) = lim s ∈ M r s ( x ) = lim s ∈ M,s ≥ s r s ( x ) ∈ V ⊂ U. This shows that lim M ∈M R M ( x ) = R M ∞ ( x ).(iii): Pick s ∈ (Γ ′ ) σ and x ∈ r s [ K ]. It is not difficult to observe that thereexists an increasing sequence { s n } in Γ ′ with sup n s n = s and so we obtain R Γ ′ ( x ) = lim n ∈ ω R Γ ′ ( r s n ( x )) = lim n ∈ ω lim t ∈ Γ ′ ,t ≥ s n r t ( r s n ( x )) = lim n ∈ ω r s n ( x ) = x. (iv): First, we claim that for every x ∈ R Γ ′ [ K ] we have lim s ∈ (Γ ′ ) σ r s ( x ) = x .Fix x ∈ R Γ ′ [ K ] and let U be a neighborhood of x . Pick an open set V with x ∈ V ⊂ V ⊂ U . Since ( r s ( x )) s ∈ Γ ′ converges to x , there exists s ∈ Γ ′ suchthat r s ( x ) ∈ V for every s ∈ Γ ′ with s ≥ s . Moreover, for every t ∈ (Γ ′ ) σ with t ≥ s there exists an increasing sequence ( s n ) n<ω ⊂ Γ ′ with sup s n = t and so we have r t ( x ) ∈ V ⊂ U . Since U was an arbitrary neighbourhood of x , this proves the claim.Thus, using (iii) and the definition of the retractional skeleton, we observethat s ′ := ( r s | R M [ K ] ) s ∈ (Γ ′ ) σ is a retractional skeleton on R Γ ′ [ K ] with D ( s ′ ) = S s ∈ (Γ ′ ) σ r s [ R Γ ′ [ K ]] ⊂ D ( s ) ∩ R Γ ′ [ K ]. On the other hand, since D ( s ) isFr´echet-Urysohn (see [21, Theorem 32]), for every x ∈ D ( s ) ∩ R Γ ′ [ K ] thereis a sequence ( s n ) n ∈ ω in Γ ′ with r s n ( x ) → x and therefore x ∈ D ( s ′ ), because D ( s ′ ) is a countably closed set. Thus, we have that D ( s ′ ) = D ( s ) ∩ R Γ ′ [ K ].(v): If ( r s ) s ∈ Γ is commutative, then for every s ∈ Γ and x ∈ K we have R Γ ′ ( r s ( x )) = lim t ∈ Γ ′ r t ( r s ( x )) = r s (lim t ∈ Γ ′ r t ( x )) ∈ D ( s ) , which implies R Γ ′ [ D ( s )] ⊂ D ( s ) and so R Γ ′ [ D ( s )] = D ( s ) ∩ R Γ ′ [ K ]. (cid:3) HARACTERIZATION OF EBERLEIN COMPACTA 7 Retractions associated to suitable models
The most important results concerning projectional skeletons were origi-nally proved in [21] using the so-called “method of suitable countable mod-els” which replaces inductive constructions by “suitable countable models”.The presentation of this method was further simplified in [6] and later it wasalso used in the context of spaces admiting retractional skeletons, see e.g. [7]or [10]. Here we further generalize and deeply investigate this method. Themain difference of our approach is that we do not consider only countablemodels. The main outcome of this section is that for every (not necessarilycountable) suitable model we can define a canonical retraction associated tothis model. Those canonical retractions will be deeply used in the remainderof the paper.Properties of retractions associated to suitable models are summarizedin Theorem 15 and, consequently, in Proposition 17 we obtain a continu-ous chain of retractions associated to suitable models with very pleasantproperties. As an example of an application we show in Theorem 21 thatwe may in a certain way combine properties of countably many retractionalskeletons.3.1.
Preliminaries.
Here we settle the notation and give some basic ob-servations concerning suitable models. We refer the interested reader to [6]and [10], where more details about this method may be found (warning:in [6, 10] only countable models were considered, while here we considersuitable models which are not necessarily countable).Any formula in the set theory can be written using symbols ∈ , = , ∧ , ∨ , ¬ , → , ↔ , ∃ , ( , ) , [ , ] and symbols for variables. On the other hand, it would be verylaborious and pointless to use only the basic language of the set theory. Forexample, we often write x < y and we know, that in fact this is a shortcutfor a formula ϕ ( x, y, < ) with all free variables shown. Thus, in what followswe will use this extended language of the set theory as we are used to, havingin mind that the formulas we work with are actually sequences of symbolsfrom the list mentioned above.Let N be a fixed set and φ be a formula. Then the relativization of φ to N is the formula φ N which is obtained from φ by replacing each quantifierof the form “ ∃ x ” by “ ∃ x ∈ N ” (and if we extend our language of set theoryby the symbol “ ∀ ” then we replace also each quantifier of the form “ ∀ x ” by“ ∀ x ∈ N ”).If φ ( x , . . . , x n ) is a formula with all free variables shown, then φ is abso-lute for N if ∀ a , . . . , a n ∈ N ( φ N ( a , . . . , a n ) ↔ φ ( a , . . . , a n )) . Definition 3.
Let Φ be a finite list of formulas and X be any set. Let M ⊃ X be a set such that each φ from Φ is absolute for M . Then wesay that M is a suitable model for Φ containing X . This is denoted by M ≺ (Φ; X ). C. CORREA, M. C ´UTH, AND J. SOMAGLIA
Note that suitable models do exist.
Theorem 4 (see Theorem IV.7.8 in [24]) . Let Φ be a finite list of formulasand X be any set. Then there exists a set R such that R ≺ (Φ; X ) and | R | ≤ max( ω, | X | )) and moreover, for every countable set Z ⊂ R thereexists M ⊂ R such that M ≺ (Φ; Z ) and M is countable. The fact that certain formula is absolute for M will always be used exclu-sively in order to satisfy the assumption of the following lemma. Using thislemma we can force the model M to contain all the needed objects created(uniquely) from elements of M . We give here the well-known proof for theconvenience of the reader. Lemma 5.
Let φ ( y, x , . . . , x n ) be a formula with all free variables shownand let M be a set that is absolute for φ and for ∃ yφ ( y, x , . . . , x n ) . If a , . . . , a n ∈ M are such that there exists a set u satisfying φ ( u, a , . . . , a n ) ,then there exists a set v ∈ M satisfying φ ( v, a , . . . , a n ) . Moreover, if thereexists a unique set u such that φ ( u, a , . . . , a n ) , then u ∈ M .Proof. It follows from the absoluteness of the formula ∃ yφ ( y, x , . . . , x n ),that there exists v ∈ M such that φ M ( v, a , . . . , a n ). Therefore the abso-luteness of the formula φ ( y, x , . . . , x n ) implies that φ ( v, a , . . . , a n ) holds.Moreover, if u is the only set such that φ ( u, a , . . . , a n ), then v = u and thus u ∈ M . (cid:3) Convention 6.
Whenever we say “ for any suitable model M (the followingholds . . . ) ” we mean that “ there exists a finite list of formulas Φ and acountable set Y such that for every M ≺ (Φ; Y ) (the following holds . . . ) ”.If M is a suitable model and h X, τ i is a topological space (or is h X, d i ametric space or is h X, + , · , k · ki a normed linear space) then we say that M contains X if h X, τ i ∈ M , h X, d i ∈ M and h X, + , · , k · ki ∈ M , respectively.The following summarizes certain easy observations. For the proofs werefer the reader to [6, Sections 2 and 3], where it is assumed that M iscountable but this fact is not used in proofs. Lemma 7.
For any suitable model M the following holds:(1) Q , ω, R ∈ M and M contains the usual operations and relations on R .(2) For every function f ∈ M we have Dom f ∈ M , Rng f ∈ M and f [ M ∩ Dom f ] ⊂ M .(3) For every finite set A we have A ∈ M if and only if A ⊂ M .(4) For every countable set A ∈ M we have A ⊂ M . Moreover, if κ ∈ M is a cardinal and κ ⊂ M then for every A ∈ M with | A | ≤ κ we have A ⊂ M .(5) For every natural number n > and sets a , . . . , a n we have { a , . . . , a n } ⊂ M if and only if h a , . . . , a n i ∈ M .(6) If A, B ∈ M , then A ∩ B ∈ M , B \ A ∈ M and A ∪ B ∈ M . HARACTERIZATION OF EBERLEIN COMPACTA 9 (7) If M contains a normed linear space X , then X ∩ M is a linearsubspace of X . Some more easy observations are summarized in the following.
Lemma 8.
For any suitable model M the following holds:(1) If (Γ , ≤ ) is up-directed and (Γ , ≤ ) ∈ M , then Γ ∩ M is up-directed.(2) If f, g ∈ M are functions and f ◦ g is well-defined, then f ◦ g ∈ M .(3) If f ∈ M is a function which is one-to-one then f − ∈ M .(4) If f ∈ M is a function and X ∈ M is a subset of Dom f , then f [ M ∩ X ] = M ∩ f [ X ] .(5) If A and B are sets and A, B ∈ M , then B A ∈ M and A × B ∈ M .(6) For every set I ∈ M and X ⊂ R I with X ∈ M we have π ∈ M ,where π : I → R X is the mapping given for i ∈ I and x ∈ X as π ( i )( x ) := x ( i ) .(7) Let X ⊂ Σ( I ) be such that I ∈ M . Then suppt( x ) ⊂ M for every x ∈ X ∩ M .(8) If ( X, τ ) is a topological space with { X, τ } ⊂ M , then {C ( X ) , + , · , ⊗} ⊂ M (where · is a multiplication by real numbers and ⊗ pointwise mul-tiplication of functions). Moreover, if X is a compact space then M contains the normed linear space C ( X ) , C ( X ) ∩ M is a closedsubalgebra of C ( X ) and ∈ C ( X ) ∩ M .(9) If ( K, τ ) is a compact space, A ⊂ C ( K ) separates the points of K and { K, τ,
A} ⊂ M , then alg(( A ∪ { } ) ∩ M ) = C ( K ) ∩ M .(10) If ( K, τ ) is a compact space , K ′ ⊂ K is closed and metrizablewith { K ′ , τ, K } ∈ M then C ( K ) ∩ M separates the points of K ′ and K ′ ⊂ K ′ ∩ M .(11) If ( K, τ ) is a compact space, D ⊂ K a dense subset with { K, D, τ } ⊂ M and f ∈ C ( K ) ∩ M , then k f k = k f | D ∩ M k Proof.
Let S and Φ be the countable set and the list of formulas from thestatement of Lemma 7, where Φ is enriched by formulas (and their subfor-mulas) marked by ( ∗ ) in the proof below. Let M ≺ (Φ; S ). Then M satisfies(1), (2), (3), (5), and (6). Indeed those items follow easily using Lemma 5and the absoluteness of the following formulas (and their subformulas) ∀ u, v ∈ Γ ∃ w ∈ Γ w ≥ u, v, ( ∗ ) ∃ h ( h = f ◦ g ) . ( ∗ ) ∃ h ( h = f − ) . ( ∗ ) ∃ W ( W = B A ) . ( ∗ ) ∃ W ( W = B × A ) . ( ∗ ) ∃ π ∈ ( R X ) I ( ∀ i ∈ I ∀ x ∈ X : π ( i )( x ) = x ( i )) . ( ∗ ) (4): By Lemma 7 (2), we have that f [ M ∩ Dom f ] ⊂ M so in particular f [ M ∩ X ] ⊂ M ∩ f [ X ]. For the other inclusion pick x ∈ f [ X ] ∩ M . UsingLemma 5 and the absoluteness of the following formula (and its subformulas) ∃ y ∈ X ( f ( y ) = x ) , ( ∗ )there exists y ∈ M ∩ X with f ( y ) = x and so x ∈ f [ M ∩ X ].(7): Pick x ∈ X ∩ M . Using Lemma 5 and absoluteness of the followingformula (and its subformulas) ∃ D ⊂ I ( i ∈ D ⇔ x ( i ) = 0) , ( ∗ )we obtain that suppt( x ) ∈ M . Since suppt( x ) is a countable set, by Lemma 7(4) we obtain that suppt( x ) ⊂ M .(8): Using Lemma 5 and absoluteness of the following formulas (and theirsubformulas) ∃C ( X ) ∈ R X ( ∀ f ∈ R X : f ∈ C ( X ) ⇔ f is continuous) , ( ∗ ) ∃ + ∈ C ( X ) C ( X ) ×C ( X ) ( ∀ f, g ∈ C ( X ) ∀ x ∈ X : +( f, g )( x ) = f ( x ) + g ( x )) , ( ∗ ) ∃· ∈ C ( X ) R ×C ( X ) ( ∀ α ∈ R ∀ f ∈ C ( X ) ∀ x ∈ X : · ( α, f )( x ) = αf ( x )) , ( ∗ ) ∃⊗ ∈ C ( X ) C ( X ) ×C ( X ) ( ∀ f, g ∈ C ( X ) ∀ x ∈ X : ⊗ ( f, g )( x ) = f ( x ) g ( x )) , ( ∗ )we obtain that C ( X ) ∈ M and that { + , · , ⊗} ⊂ M . Morevoer, if X is acompact space, then using Lemma 5 and the absoluteness of the followingformula (and its subformulas) ∃k · k ∞ ∈ R C ( X ) ( ∀ f ∈ C ( X ) : k · k ( f ) = sup x ∈ X | f ( x ) | ) , ( ∗ )we obtain that M contains the normed linear space C ( X ). Thus, by Lemma 7(7), C ( X ) ∩ M is a closed subspace of C ( X ) and, since ⊗ ∈ M , C ( X ) ∩ M is closed under multiplication and therefore C ( X ) ∩ M is closed undermultiplication as well. Finally, using Lemma 5 and absoluteness of thefollowing formula (and its subformulas) ∃ f ∈ C ( X ) ( ∀ x ∈ X f ( x ) = 1) , ( ∗ )we obtain that 1 ∈ C ( X ) ∩ M .(9): By (8), C ( K ) ∩ M is a closed subalgebra of C ( K ) that contains ( A ∪{ } ) ∩ M , so we have alg(( A ∪ { } ) ∩ M ) ⊂ C ( K ) ∩ M . For the other in-clusion, pick f ∈ C ( K ) ∩ M . By Lemma 5 and absoluteness of the followingformula (and its subformulas) ∃ A ⊂ A ( A is countable and f ∈ alg( A ∪ { } )) , ( ∗ )there is a countable set A ⊂ A with A ∈ M and f ∈ alg( A ∪ { } ). ByLemma 7 (4), we have that A ⊂ A ∩ M . Therefore, using that 1 ∈ M , weobtain alg(( A ∪ { } ) ∩ M ) = alg(( A ∩ M ) ∪ { } ) ⊃ C ( K ) ∩ M .
HARACTERIZATION OF EBERLEIN COMPACTA 11 (10): By (8), Lemma 5 and the absoluteness of the following formula (andits subformulas) ∃ A ⊂ C ( K ) ( A is countable and separates the points of K ′ ) , ( ∗ )there is a countable set A ⊂ C ( K ) with A ∈ M which separates the points of K ′ . By Lemma 7 (4), we have that A ⊂ C ( K ) ∩ M so C ( K ) ∩ M separates thepoints of K ′ . Therefore, since by (8) the set C ( K ) ∩ M is a closed algebracontaining constant functions, Stone-Weierstrass theorem ensures that theset { f | K ′ : f ∈ C ( K ) ∩ M } is dense in C ( K ′ ), which implies that { f | K ′ : f ∈C ( K ) ∩ M } is dense in C ( K ′ ) and therefore { f − ( − / , / ∩ K ′ : f ∈ C ( K ) ∩ M } is an open basis of K ′ . Moreover, for every f ∈ C ( K ) ∩ M using Lemma 5and the absoluteness of the following formula (and its subformulas) ∃ x ∈ f − ( − / , / ∩ K ′ , ( ∗ )we have that f − ( − / , / ∩ ( K ′ ∩ M ) = ∅ for every f ∈ C ( K ) ∩ M andtherefore the set K ′ ∩ M is dense in K ′ .(11): Since D ⊂ K is a dense set, we have that k f k = k f | D k . It followsfrom (8) that k · k ∈ M and so k f k ∈ M . Therefore, using Lemma 5 andthe absoluteness of the following formula (and its subformulas) ∀ n ∈ ω ∃ x ∈ D ( k f k − /n < | f ( x ) | < k f k + 1 /n ) , ( ∗ )we obtain that for every n ∈ ω , there exists x n ∈ D ∩ M such that | f ( x n ) | →k f k . (cid:3) Retractions associated to suitable models.
Here we show thatin a compact space with a retractional skeleton, for every suitable modelthere is a canonical retraction associated to it (see Definition 12). Themain outcome of this subsection is Theorem 15, where the properties of acanonical retraction are summarized.Lemma 9 and Lemma 10 are inspired by [10, Lemma 4.7], where somethingsimilar was proved for suitable models which are countable.
Lemma 9.
For every suitable model M the following holds: Let X be a setand A ⊂ R X such that { X } ∪ A ⊂ M . Consider the mapping q M : X → R A defined for x ∈ X as q M ( x )( f ) := f ( x ) , f ∈ A . Then for every B ⊂ X with B ∈ M we have q M [ B ] ⊂ q M [ B ∩ M ] .Proof. In this proof we will use the identification of any n ∈ ω with the set { , ..., n − } . Further, denote by B the set of all the open intervals withrational endpoints and by B <ω the set of all the functions whose domain issome n ∈ ω and whose values are in B .Let S be the countable set from the statement of Lemma 7 enriched by {B , B <ω } and let Φ be the list of formulas from the statement of Lemma 7enriched by formulas (and their subformulas) marked by ( ∗ ) in the proofbelow. Let M ≺ (Φ; S ∪ { X } ∪ A ).Fix B ⊂ X with B ∈ M , a point x ∈ B and a basic neighborhood ofa point q M ( x ); that is, let us pick finitely many functions F ⊂ A and a sequence of rational intervals such that f ( x ) ∈ I f , f ∈ F and consider theneighborhood N := { y ∈ R A : y ( f ) ∈ I f for every f ∈ F } . By Lemma 7 (3), we have that F ∈ M and by absoluteness of the formula ∃ n ∈ ω ∃ η ( η is a bijection between n and F ) , ( ∗ )and its subformulas, there is n ∈ ω and a bijection η ∈ M between n and F .Let us further define the mapping ξ : n → B by ξ ( i ) = I η ( i ) . Since ξ ∈ B <ω ∈ M , it follows from Lemma 7 (4) that ξ ∈ M . By Lemma 5 andthe absoluteness of the formula (and its subformulas) ∃ x ∈ B ( ∀ i < n : η ( i )( x ) ∈ ξ ( i )) , ( ∗ )there is a point x ∈ B ∩ M such that q M ( x ) ∈ N ; hence, q M [ B ∩ M ] isdense in q M [ B ]. (cid:3) Lemma 10.
For every suitable model M the following holds: Let ( K, τ ) bea compact space and D ⊂ K be a dense subset with { K, D, τ } ⊂ M .If C ( K ) ∩ M separates the points of D ∩ M , then there exists a uniqueretraction r M : K → D ∩ M such that f = f ◦ r M , for every f ∈ C ( K ) ∩ M .Moreover, in this case(1) for every x ∈ K and A ⊂ C ( K ) separating the points of K with A ∈ M , r M ( x ) is the unique point from D ∩ M satisfying f ( r M ( x )) = f ( x ) , for every f ∈ A ∩ M .(2) if B ⊂ D and B ∈ M , then r M [ B ] = B ∩ M .Proof. Let S and Φ be the union of sets and the lists of formulas from thestatements of Lemma 7, Lemma 8 and Lemma 9. Let M ≺ (Φ; S ∪{ K, D, τ } )be such that C ( K ) ∩ M separates the points of D ∩ M . By Lemma 8 (8),we have that C ( K ) ∈ M .Let us consider the mapping q M : K → R C ( K ) ∩ M given by q M ( x ) =( f ( x )) f ∈C ( K ) ∩ M , x ∈ K . Then q M is continuous and, by the assump-tion, q M | D ∩ M is one-to-one; hence, q M | D ∩ M is a homeomorphic embedding.Moreover, whenever B ⊂ D is such that B ∈ M , then by Lemma 9 we have q M [ B ∩ M ] ⊃ q M [ B ] which implies q M [ B ] = q M [ B ∩ M ].Now, put r M := ( q M | D ∩ M ) − ◦ q M . Then it is a continuous retractionwith r M [ K ] = D ∩ M . Moreover, for every x ∈ Kr M ( x ) = ( q M | D ∩ M ) − ◦ q M ( x ) = y, where y ∈ K is the unique point such that y ∈ D ∩ M and g ( y ) = g ( x ), forevery g ∈ C ( K ) ∩ M . Hence, for f ∈ C ( K ) ∩ M we have f ( r M ( x )) = f ( y ) = f ( x ) . In order to see that r M is unique, let us consider another retraction r ′ : K → D ∩ M satisfying that f = f ◦ r ′ for every f ∈ C ( K ) ∩ M . Then,for every x ∈ K , and every f ∈ C ( K ) ∩ M we have f ( r M ( x )) = f ( x ) = f ( r ′ ( x )); hence, since C ( K ) ∩ M separates the points of r M [ K ], it holds that HARACTERIZATION OF EBERLEIN COMPACTA 13 r M ( x ) = r ′ ( x ). Since x ∈ K was arbitrary, we have r M = r ′ . Moreover,given y ∈ D ∩ M such that f ( y ) = f ( x ) for every f ∈ A ∩ M , where A ⊂ C ( K ) is a set separating the points of K with A ∈ M , we obtain that f ( y ) = f ( x ) for f ∈ alg(( A ∪ { } ) ∩ M ) = C ( K ) ∩ M (the last equalityfollows from Lemma 8 (9)) and so y = r M ( x ).Finally, if B ⊂ D is such that B ∈ M then by the above we have q M [ B ] = q M [ B ∩ M ] and so r M [ B ] = r M [ B ∩ M ] = B ∩ M . (cid:3) Let us note that a compact space K admits a retractional skeleton ifand only if there exists a dense set D ⊂ K such that for every suitablemodel M which is moreover countable, the set C ( K ) ∩ M separates thepoints of D ∩ M , see e.g. [7, Theorem 4.9] or [18, Theorem 19.16] (thatis, the assumption of Lemma 10 is satisfied for suitable models which arecountable). The following shows that we do not need to assume countabilityof the model. Proposition 11.
For every suitable model M the following holds: If ( K, τ ) is a compact space and D ⊂ K is a subset of a set induced by a retractionalskeleton with { D, K, τ } ⊂ M , then C ( K ) ∩ M separates the points of D ∩ M .Proof. Let S be the union of sets from the statements of Lemma 7 andLemma 8 and let Φ be the union of lists of formulas from the statementsof Lemma 7 and Lemma 8 enriched by formulas (and their subformulas)marked by ( ∗ ) in the proof below. Let M ≺ (Φ; S ∪ { K, D, τ } ).By Lemma 5 and the absoluteness of the following formula (and its sub-formulas) ∃ Γ ∃ ≤ ∃ r (cid:0) D is a subset of a set induced bythe retractional skeleton { r ( s ) : s ∈ Γ } (cid:1) , ( ∗ )there exist Γ , ≤ , r ∈ M such that { r ( s ) : s ∈ Γ } is a retractional skeleton on K inducing a set containing D . For s ∈ Γ we will write below r s instead of r ( s ). By Lemma 8 (1), the set Γ ∩ M is up-directed. Hence by Lemma 2there exists a continuous retraction R M : K → K defined by R M ( x ) :=lim s ∈ Γ ∩ M r s ( x ), for every x ∈ K . Using the absoluteness of the followingformula (and its subformulas) ∀ u ∈ D ∃ s ∈ Γ u ∈ r s [ K ] , ( ∗ )we obtain that D ∩ M ⊂ R M [ K ] and therefore D ∩ M ⊂ R M [ K ]. Now fix x, y ∈ D ∩ M with x = y . Since x = lim s ∈ Γ ∩ M r s ( x ) and y = lim s ∈ Γ ∩ M r s ( y )there exists s ∈ Γ ∩ M such that r s ( x ) = r s ( y ). By Lemma 7 (2), we havethat r ( s ) = r s ∈ M and r s [ K ] ∈ M . Thus, by Lemma 8 (10), there exists f ∈ C ( K ) ∩ M such that f ( r s ( x )) = f ( r s ( y )). Now using Lemma 8 (2),we obtain that g = f ◦ r s ∈ C ( K ) ∩ M and g ( x ) = g ( y ). Thus, C ( K ) ∩ M separates the points of D ∩ M (cid:3) The retraction constructed in Lemma 10 (whose assumption is satisfiedby Proposition 11 in compact spaces admitting a retractional skeleton) willbe the key to our considerations. Let us give it a name.
Definition 12.
Let K be a compact space and let D ⊂ K be a dense subsetthat is contained in the set induced by a retractional skeleton. Given a set M , we say that r M is the canonical retraction associated to M , K and D ifit is the unique retraction on K satisfying r M [ K ] = D ∩ M and f = f ◦ r M ,for every f ∈ C ( K ) ∩ M . We say that a set M admits canonical retraction if there exists the canonical retraction associated to M , K and D .In the case when D = K we say that M admits canonical retraction r M associated to M and K .The properties of canonical retractions associated to suitable models aresummarized in Theorem 15. We need two lemmas first. Lemma 13.
For every suitable model M the following holds: Let ( K, τ ) bea compact space and A ⊂ C ( K ) be a set separating the points of K with {A , K, τ } ⊂ M . Then for every compact set K ′ ⊂ K , A ∩ M separates thepoints of K ′ if and only if C ( K ) ∩ M separates the points of K ′ .Proof. Let S and Φ be the set and the list of formulas from the statementsof Lemma 7 and Lemma 8. Let M ≺ (Φ; S ∪ { K, τ, A} ). In order to get acontradiction, let us assume that C ( K ) ∩ M separates the points of K ′ but A∩ M does not separate the points of K ′ . Then also alg(( A ∩ M ) ∪ { } ) doesnot separate the points of K ′ (because if there are x = y with f ( x ) = f ( y )for every f ∈ A ∩ M , then also g ( x ) = g ( y ) for every g of the form g = a + P ni =1 a i Π mj =1 f i,j ). But this is a contradiction, because using Lemma 8(8) and (9) we conclude that alg(( A ∩ M ) ∪ { } ) = C ( K ) ∩ M . (cid:3) Lemma 14.
For every suitable model M the following holds: Let ( K, τ ) be acompact space and let D ⊂ K be a set induced by a retractional skeleton suchthat { K, D, τ } ⊂ M . Then the mapping Φ : C ( K ) ∩ M → C ( D ∩ M ) definedby Φ( f ) := f | D ∩ M , for every f ∈ C ( K ) ∩ M , is a surjective isometry.Proof. Let S and Φ be the union of countable sets and finite lists of formulasfrom the statements of Lemma 8 and Proposition 11. Let M ≺ (Φ; S ∪{ K, D, τ } ). By Lemma 8 (11), we have that k f k = k f | D ∩ M k , for every f ∈ C ( K ) ∩ M , so the mapping Φ | C ( K ) ∩ M is an isometry which implies thatΦ is also an isometry. It remains to show that it is surjective. By Lemma 8(8), C ( K ) ∩ M is a closed subalgebra of C ( K ) and so the image of Φ isa closed subalgebra of C ( D ∩ M ) which, by Proposition 11 separates thepoints of D ∩ M . Therefore, it follows from Stone-Weierstrass theorem thatΦ[ C ( K ) ∩ M ] = C ( D ∩ M ). (cid:3) Theorem 15.
For every suitable model M the following holds: Let ( K, τ ) be a compact space and let D ⊂ K be a dense subset that is contained in theset induced by a retractional skeleton with { K, D, τ } ⊂ M . Then there exists HARACTERIZATION OF EBERLEIN COMPACTA 15 a unique retraction r M : K → D ∩ M with r M [ K ] = D ∩ M and f = f ◦ r M ,for every f ∈ C ( K ) ∩ M . Moreover, for this retraction r M the followingholds:(i) Whenever A ⊂ C ( K ) separates the points of K and A ∈ M , thenfor every x ∈ K , r M ( x ) is the unique point from D ∩ M satisfying f ( r M ( x )) = f ( x ) , for every f ∈ A ∩ M .(ii) Whenever (Γ , ≤ ) is up-directed and σ -complete and r : Γ → C ( K, K ) is a mapping such that s = { r ( s ) : s ∈ Γ } is a retractional skeletonon K inducing D and { r, Γ , ≤} ⊂ M , then the following holds.(a) For every s ∈ Γ ∩ M , r ( s )[ K ] ⊂ D ∩ M .(b) r M ( x ) = lim s ∈ Γ ∩ M r ( s )( x ) , x ∈ K. (c) If M is countable, then r M = r ( s ) for s = sup Γ ∩ M .(d) { r ( s ) | r M [ K ] : s ∈ (Γ ∩ M ) σ } is a retractional skeleton on r M [ K ] with induced set D ∩ r M [ K ] .(e) If s is commutative, then r M [ D ] = D ∩ r M [ K ] .(iii) Whenever h : K → L is a surjective homeomorphism with h ∈ M ,then r := h ◦ r M ◦ h − is the unique retraction on L such that r [ L ] = h [ D ] ∩ M and f ◦ r = f , for every f ∈ C ( L ) ∩ M .(iv) w( r M [ K ]) ≤ | M | .Proof. Denote by B the set of all open intervals with rational endpoints.Let S be the union of sets from the statements of Lemma 7, Lemma 8,Lemma 10, Lemma 14 and Proposition 11 and let Φ be the union of lists offormulas from the statements of Lemma 7, Lemma 8, Lemma 10, Lemma 14and Proposition 11 enriched by the formula (and its subformulas) markedby ( ∗ ) in the proof below. Let M ≺ (Φ; S ∪ { K, D, τ } ). It follows directlyfrom Proposition 11 and Lemma 10 that the retraction r M exists and thatit satisfies (i).Let { r, Γ , ≤} ⊂ M be as in (ii). For s ∈ Γ ∩ M , by Lemma 7 (2) wehave that r ( s )[ K ] ∈ M and thus Lemma 8 (10) ensures that r ( s )[ K ] ⊂ r ( s )[ K ] ∩ M ⊂ D ∩ M , so (a) holds. By Lemma 8 (1), Γ ∩ M is up-directedand so, using Lemma 2 the limit lim s ∈ Γ ∩ M r ( s )( x ) =: R M ( x ) exists for every x ∈ K .We claim that R M = r M . Note that due to the uniqueness of r M , itis enough to show that R M [ K ] ⊂ D ∩ M and that f ◦ R M = f , for every f ∈ C ( K ) ∩ M . By the definition of R M , using (a), we observe that R M [ K ] ⊂ D ∩ M . Moreover, for every x ∈ D ∩ M , by Lemma 5 and the absolutenessof the following formula (and its subformulas) ∃ s ∈ Γ (cid:0) r s ( x ) = x (cid:1) , ( ∗ )there is s ∈ Γ ∩ M with r s ( x ) = x and so x ∈ R M [ K ]. Thus, we have that R M [ K ] = D ∩ M . Pick f ∈ C ( K ) ∩ M . Since ( r s ) s ∈ Γ ∩ M converges pointwiseto R M , using [20, Lemma 5.2] we conclude that ( f ◦ r s ) s ∈ Γ ∩ M converges in norm to f ◦ R M . Therefore f ◦ R M ∈ C ( K ) ∩ M , since it follows fromLemma 7 (2) and Lemma 8 (2) that f ◦ r s ∈ C ( K ) ∩ M , for every s ∈ Γ ∩ M .Thus f and f ◦ R M are two functions from C ( K ) ∩ M which have the samevalues on D ∩ M and so, by Lemma 14, f = f ◦ R M . This proves the claimand establishes (b) which, using Lemma 2, implies (c), (d) and (e).The proof of (iii) is easy once we realize that by Lemma 8 (4) we have that h [ D ∩ M ] = h [ D ] ∩ M and that f ◦ h ∈ C ( K ) ∩ M , for every f ∈ C ( L ) ∩ M .We omit the straightforward details.Finally, (iv) follows from Proposition 11, since r M [ K ] = D ∩ M . (cid:3) Families of canonical retractions.
Here we study families of canoni-cal retractions associated to suitable models. Those are more-or-less straight-forward consequences of Theorem 15. The most important for what followsis Proposition 17 which will be repeatedly used further.
Lemma 16.
There are a countable set S and a finite list of formulas Φ such that the following holds: Let ( K, τ ) be a compact space, (Γ , ≤ ) be anup-directed set, r : Γ → C ( K, K ) be a mapping such that s := { r ( s ) : s ∈ Γ } is a retractional skeleton on K and let D ⊂ D ( s ) be dense in K . Put S ′ = S ∪ { K, τ, D, Γ , ≤ , r } . Then every M ≺ (Φ; S ′ ) admits the canonicalretraction r M associated to M , K and D .Moreover, we have the following.(1) If M, N ≺ (Φ; S ′ ) and M ⊂ N then r M ◦ r N = r N ◦ r M = r M .(2) Let M be an up-directed set with M ≺ (Φ; S ′ ) , for every M ∈ M andlet M ∞ := S M ∈M M . Then M ∞ ≺ (Φ; S ′ ) and lim M ∈M r M ( x ) = r M ∞ ( x ) , x ∈ K .(3) If U is a basis of τ , M is an up-directed set with M ≺ (Φ; S ′ ) , forevery M ∈ M and U ⊂ S M ∈M M , then lim M ∈M r M ( x ) = x , forevery x ∈ K .Proof. The existence of S and Φ follows from Theorem 15. Let us provethe moreover part using the additional properties of canonical retractionsestablished in Theorem 15.(1) Since r M [ K ] = D ∩ M ⊂ D ∩ N = r N [ K ], we have r M = r N ◦ r M .Moreover, for every f ∈ C ( K ) ∩ M ⊂ C ( K ) ∩ N we have f ( r M ( x )) = f ( x ) = f ( r N ( x )) = f ( r M ( r N ( x ))) , x ∈ K and so r M ( x ) = r M ( r N ( x )), for every x ∈ K .(2) Since M is up-directed, it follows from [6, Lemma 2.1] and Lemma 5that M ∞ ≺ ( S ′ ; Φ). Now, combining Theorem 15 (ii) with Lemma 8 (1) andLemma 2 (ii) we obtain that lim M ∈M r M ( x ) = r M ∞ ( x ), x ∈ K .(3) Pick x ∈ K , U ∈ U such that x ∈ U and find V ∈ U with x ∈ V ⊂ V ⊂ U .Since V ∈ U and U ⊂ S M ∈M M , there exists M ∈ M such that V ∈ M .Now fix M ∈ M with M ⊂ M . It follows from Lemma 7 (6) that V ∩ D ∈ M . Therefore Lemma 10 ensures that r M ( x ) ∈ r M [ V ∩ D ] ⊂ V ∩ D ⊂ U ,since x ∈ V ∩ D . (cid:3) HARACTERIZATION OF EBERLEIN COMPACTA 17
Proposition 17.
There exist a countable set S and a finite list of formulas Φ such that the following holds:Let ( K, τ ) be a compact space, (Γ , ≤ ) be an up-directed set and r : Γ →C ( K, K ) be a mapping such that s = { r ( s ) : s ∈ Γ } is a retractional skeletonon K . Let κ := w( K ) and U : κ → τ be such that {U ( i ) : i < κ } is an openbasis of τ . Put S ′ := S ∪ { K, D ( s ) , Γ , ≤ , r, τ, U } . Let ( M α ) α ≤ κ be a sequenceof sets satisfying(Ra) M α ≺ (Φ; S ′ ) , for every α ∈ [0 , κ ] ,(Rb) | M α | ≤ max( ω, | α | ) , for every α ∈ [0 , κ ] ,(Rc) M α +1 ⊃ M α ∪ { α } , for every α ∈ [0 , κ ) ,(Rd) M α = S β<α M β , if α ∈ (0 , κ ] is a limit ordinal.Then for every α ∈ [0 , κ ] there exists a canonical retraction r α associated to M α , K and D ( s ) and the following holds.(R1) For every α < β , we have that r α ◦ r β = r β ◦ r α = r α .(R2) r α ( x ) → x , for every x ∈ K .(R3) Let α ≤ κ , let η : [0 , α ) → κ be an increasing function and let ξ ≤ κ be a limit ordinal with sup β<α η ( β ) = ξ . Then lim β<α r η ( β ) ( x ) = r ξ ( x ) , for every x ∈ K .(R4) r κ = id .(R5) For every α ∈ [0 , κ ] , we have that w( r α [ K ]) ≤ max ( ω, | α | ) and ( r s | r α [ K ] ) s ∈ (Γ ∩ M α ) σ is a retractional skeleton on r α [ K ] with inducedset D ( s ) ∩ r α [ K ] .(R6) If A is a closed subset of C ( K ) and f ◦ r s ∈ A , for every f ∈ A and s ∈ Γ , then { f ◦ r α : f ∈ A , α ≤ κ } ⊂ A .(R7) If x, y are distinct points in K then β := min { α < κ : r α ( x ) = r α ( y ) } exists and it is a successor ordinal or β = 0 .(R8) For every α ≤ κ , the set Γ ∩ M α is up-directed and r α ( x ) = lim s ∈ ( M α ∩ Γ) r s ( x ) ,for every x ∈ K .(R9) For every α ≤ κ and t ∈ (Γ ∩ M α ) σ , it holds that r t ◦ r α = r α ◦ r t .(R10) Let A ⊂ C ( K ) be a set that separates the points of K and α < κ . If A ∈ M α , then A ∩ M α separates the points of r α [ K ] and f ◦ r α = f ,for every f ∈ A ∩ M α .Moreover, if s is full or commutative, then r α [ D ( s )] ⊂ D ( s ) , for every α ∈ [0 , κ ] and if D ⊂ D ( s ) is such that r α [ D ] ⊂ D for every α ∈ [0 , κ ] , then wealso have the following.(R11) The sets { r α ( x ) : α < κ } and { α < κ : r α ( x ) = r α +1 ( x ) } are count-able, for every x ∈ D .Proof. Let S and Φ be the union of sets and lists of formulas from the state-ments of Theorem 15 and Lemma 16. Then the existence of r α , α ∈ [0 , κ ]follows from Theorem 15. Now, (R1) follows immediately from Lemma 16.(R2) and (R3) follow from Lemma 16 as well (using for (R2) the fact that {U ( i ) : i < κ } ⊂ S α<κ M α and for (R3) the fact that S β<α M η ( β ) = M ξ ).(R4) follows from (R2) and (R3) applied to η ( i ) := i , i < κ and ξ = κ . (R5) and (R10) follow from Theorem 15. For (R6) we observe that by Theo-rem 15 (ii) the net of continuous retractions ( r s ) s ∈ Γ ∩ M α converges pointwiseto the continuous retraction r α and so [20, Lemma 5.2] ensures that thenet ( f ◦ r s ) s ∈ Γ ∩ M α converges in norm to f ◦ r α , for every f ∈ C ( K ), whichimplies (R6). For (R7) we observe that by (R2) there is i < κ such that r i ( x ) = r i ( y ) so β is well defined and if β = 0 then it is a successor ordinalby (R3). For (R8) is suffices to apply Lemma 8 (1) and Theorem 15 (ii). Inorder to prove (R9), fix x ∈ K . If t ∈ Γ ∩ M α , then using (R8), we obtainthat:(1) r t (cid:0) r α ( x ) (cid:1) = lim s ∈ Γ ∩ M α ,s ≥ t r t (cid:0) r s ( x ) (cid:1) = lim s ∈ Γ ∩ M α ,s ≥ t r s (cid:0) r t ( x ) (cid:1) = r α (cid:0) r t ( x ) (cid:1) . If t ∈ (Γ ∩ M α ) σ \ (Γ ∩ M α ), then there exists an increasing sequence ( t n ) n ∈ ω of elements of Γ ∩ M α such that t = sup n ∈ ω t n . Thus it follows from equation(1) that: r t (cid:0) r α ( x ) (cid:1) = lim n ∈ ω r t n (cid:0) r α ( x ) (cid:1) = lim n ∈ ω r α (cid:0) r t n ( x ) (cid:1) = r α (cid:0) r t ( x ) (cid:1) . Moreover, if s is full then we obviously have r α [ D ( s )] = r α [ K ] ⊂ K = D ( s )and if it is commutative then Theorem 15 (ii) ensures that r α [ D ( s )] ⊂ D ( s ).(R11): Pick x ∈ D and note that in order to prove that { r α ( x ) : α < κ } is countable, it suffices to show that for every strictly increasing function η : [0 , ω ) → κ , there is ζ < ω with r η ( ζ ) ( x ) = r η ( β ) ( x ), for every ζ <β < ω . Let η : [0 , ω ) → κ be a strictly increasing function and set ξ := sup β<ω η ( β ). By (R3), we have r ξ ( x ) = lim β<ω r η ( β ) ( x ). Hence, since r ξ [ D ] ⊂ D and D has countable tightness (see [21, Theorem 32]), thereis a ζ < ω with r ξ ( x ) ∈ r η ( ζ ) [ K ]; so, for ζ ≤ β < ω , using (R1), weobtain r η ( β ) ( x ) = r η ( β ) ( r ξ ( x )) = r ξ ( x ). Finally, to conclude that the set { α < κ : r α ( x ) = r α +1 ( x ) } is countable, note that the mapping ϕ ( α ) = r α ( x ) is an injection from this set into { r α ( x ) : α < κ } . Indeed, suppose bycontradiction that there exist α, β < κ with α = β , r α ( x ) = r α +1 ( x ) and r β ( x ) = r β +1 ( x ) such that r α ( x ) = r β ( x ). Without loss of generality, wemay assume that α < β . Then applying the map r α +1 , by (R1), we obtain r α ( x ) = r α +1 ( r α ( x )) = r α +1 ( r β ( x )) = r α +1 ( x ), which is a contradiction. (cid:3) Application - passing to a subskeleton.
Here we introduce the no-tion of a (weak) subskeleton and show that for a countable family of retrac-tional skeletons inducing the same set there is a common weak subskeleton,see Theorem 21.
Definition 18.
Let K be a compact space and let s = ( r s ) s ∈ Γ be a retrac-tional skeleton on K . We say that ( r s ) s ∈ Γ ′ is a subskeleton of s , if Γ ′ ⊂ Γ isa σ -complete and cofinal subset.It is easy to see that every subskeleton is a retractional skeleton. Definition 19.
Let ( r s ) s ∈ Γ be a retractional skeleton on a compact space K . We say that ( R i ) i ∈ Λ is a weak subskeleton of ( r s ) s ∈ Γ if ( R i ) i ∈ Λ is aretractional skeleton on K and there exists a mapping φ : Λ → Γ such that
HARACTERIZATION OF EBERLEIN COMPACTA 19 • R i = r φ ( i ) , for every i ∈ Λ; • φ is ω -monotone, that is, if i, j ∈ Λ with i ≤ j , then φ ( i ) ≤ φ ( j )and if ( i n ) n ∈ ω is an increasing sequence from Λ, then sup n φ ( i n ) = φ (sup n i n ); • { φ ( i ) : i ∈ Λ } is cofinal in Γ.Clearly, every subskeleton of a retractional skeleton is also a weak sub-skeleton. Some basic properties of weak subskeletons are summarized below. Fact 20.
Let ( r s ) s ∈ Γ be a retractional skeleton on a compact space K and ( R i ) i ∈ Λ be a weak subskeleton of ( r s ) s ∈ Γ . Then • ( R i ) i ∈ Λ induces the same subset as ( r s ) s ∈ Γ ; • if ( S j ) j ∈ ∆ is a weak subskeleton of ( R i ) i ∈ Λ , then it is a weak sub-skeleton of ( r s ) s ∈ Γ . The following result shows that we may concentrate properties of count-ably many retractional skeletons into one skeleton, which is moreover gen-erated by suitable models.
Theorem 21.
Let K be a compact space and let ( r ns ) s ∈ Γ n , n ∈ ω be asequence of retractional skeletons on K inducing the same set D . Thenthere exists a retractional skeleton which is a weak subskeleton of ( r ns ) s ∈ Γ n ,for every n ∈ ω .Moreover, for every countable set S and every finite list of formulas Φ ,there exists a family M consisting of countable suitable models for Φ con-taining S such that every M ∈ M admits canonical retraction r M associatedto M , K and D and ( r M ) M ∈M is a weak subskeleton of ( r ns ) s ∈ Γ n , for every n ∈ ω , where the ordering on M is given by inclusion.Proof. Let Γ n = (Γ n , ≤ n ) and r n : Γ n → C ( K, K ) be such that r n ( s ) := r ns ,for every n ∈ ω and s ∈ Γ n . Let τ be the topology on K . Let S ′ be theunion of S and the countable set from the statement of Theorem 15 enrichedby { K, D, τ, r n , Γ n , ≤ n : n ∈ ω } and let Φ ′ be the union of Φ and the list offormulas from the statement of Theorem 15. By Theorem 4, there is a set R ⊃ S ′ ∪ τ ∪ S n ∈ ω Γ n such that R ≺ (Φ ′ ; S ′ ) and for every countable set Z ⊂ R there is a countable set M ( Z ) ⊂ R satisfying M ( Z ) ≺ (Φ ′ ; Z ). Set M = { M ∈ [ R ] ω : M ≺ (Φ ′ , S ′ ) } , ordered by inclusion. The σ -completeness of M follows from [6, Lemma 2.4].To see that M is up-directed, let N , N ∈ M , then the set M ( N ∪ N ) ∈ M and satisfies N ∪ N ⊂ M ( N ∪ N ). By Theorem 15, every M ∈ M admitscanonical retraction r M associated to M , K and D . Note that for every U ∈ τ there is M ∈ M with U ∈ M (it suffices to put M = M ( { U }∪ S ′ )) andso τ ∩ ( S Λ) = τ is indeed a basis of the topology τ . Therefore it follows fromTheorem 15 (iv) and Lemma 16 that ( r M ) M ∈M is a retractional skeletonon K . Now for every n ∈ ω , let φ n : M → Γ n be the mapping defined by φ n ( M ) := sup(Γ n ∩ M ). By Lemma 15 (ii), we have that r M = r nφ n ( M ) , for every n ∈ ω . Now, let ( M k ) k ∈ ω ⊂ M be an increasing sequence, thenit is easy to see that sup k φ n ( M k ) = φ n ( M ∞ ), where M ∞ = S k ∈ ω M k . Itremains to prove that the set { φ n ( M ) : M ∈ M} is cofinal in Γ n , for each n ∈ ω . Let n ∈ ω and s ∈ Γ n , then there exists M ∈ M such that s ∈ M (it suffices to put M = M ( { s } ∪ S ′ )). Therefore φ n ( M ) = sup(Γ n ∩ M ) ≥ s .This concludes the proof. (cid:3) Valdivia embedding of compact spaces admitting acommutative skeleton
Giving a compact space K that admits a commutative retractional skele-ton s , it is known that there exists a homeomorphic embedding h : K → [ − , I such that h [ D ( s )] ⊂ Σ( I ) (that is, K is Valdivia and D ( s ) is aΣ-subset of K ). The main aim of this section is to have a very concreteand very flexible way of understanding the mapping h , which is the topichandled in Subsection 4.2, where the proof of Theorem C is given. Apartfrom Theorem C, we would like to highlight Theorem 25 which gives evena better insight for spaces of weight ω and Theorem 28 which gives a newcharacterization of Valdivia compacta using suitable models.4.1. Canonical retractions associated to suitable models in Valdiviacompact spaces.
The goal here is to obtain the following concrete descrip-tion of the canonical retractions associated to suitable models in Valdiviacompact spaces.
Lemma 22.
For every suitable model M the following holds: Let D ⊂ Σ( I ) be such that K = D is compact and { K, D, I, τ } ⊂ M (where τ is thetopology on K ). Then the mapping r : K → K defined as r ( x ) = x | I ∩ M , forevery x ∈ K , is the canonical retraction associated to M , K and D . This was in a certain sense most probably well-known for countable mod-els (see e.g. [23, Lemma 2.4]), here we show that the situation is the samefor uncountable models as well. The remainder of this subsection is more-or-less devoted to the proof of Lemma 22. We start with two preliminaryresults.
Lemma 23.
For every suitable model M the following holds: Let X ⊂ Σ( I ) be such that { X, I } ⊂ M . Then we have X ∩ M = { x | I ∩ M : x ∈ X } . Proof.
Let S and Φ be the union of sets and lists of formulas from thestatements of Lemma 7, Lemma 8 and Lemma 9 and let M ≺ (Φ; S ∪{ X, I } ).Let π : I → R X be the mapping given by π i ( x ) = x ( i ), for every x ∈ X and i ∈ I . By Lemma 7 (2) and Lemma 8 (6), we have that π ∈ M and A := π [ I ∩ M ] ⊂ M . Let q M : X → R A be the mapping from Lemma 9,that is, for x ∈ X we have q M ( x )( π i ) = x ( i ), i ∈ I ∩ M . Consider themapping φ : R A → R I given for x ∈ R A by φ ( x )( i ) := x ( π i ), if i ∈ I ∩ M HARACTERIZATION OF EBERLEIN COMPACTA 21 and φ ( x )( i ) := 0, if i ∈ I \ M . It is easy to see that φ is continuous. ByLemma 9, we have q M [ X ] ⊂ q M [ X ∩ M ] which implies that { x | I ∩ M : x ∈ X } = φ ( q M [ X ]) ⊂ φ ( q M [ X ∩ M ]) = { x | I ∩ M : x ∈ X ∩ M } . Thus, it suffices to note that for every x ∈ X ∩ M we have x | I ∩ M = x , whichfollows from the fact that the support of every x ∈ X ∩ M is contained in M , see Lemma 8 (7). (cid:3) The following is well-known. We did not find a suitable reference, but itfollows e.g. from the proof of [23, Theorem 6.1] (for the key step see also [2,Lemma 1.2]). For the convenience of the reader we show a short argumentbased on our previous considerations.
Lemma 24.
Let K ⊂ R I be a compact space and let D := Σ( I ) ∩ K bedense in K . Put Γ := { A ∈ [ I ] ≤ ω : x | A ∈ K for every x ∈ K } and for every A ∈ Γ define r A : K → K by r A ( x ) := x | A , x ∈ K . Then ( r A ) A ∈ Γ is a commutative retractional skeleton on K inducing the set D .Proof. It is obvious that each r A is a continuous retraction with r A [ K ]metrizable. For every A, B ∈ Γ, we have A ∩ B ∈ Γ and r A ◦ r B = r A ∩ B which implies that r A ◦ r B = r B ◦ r A . Having an increasing sequence ( A n ) n ∈ ω from Γ and x ∈ K , we have r A n x → x | S A n and so A ∞ := S A n ∈ Γ and r A n x → r A ∞ x . Let us now observe that for every x ∈ D there is A ∈ Γwith r A x = x . Indeed, any x ∈ D has a countable support, so it suffices tosee that for every countable E ⊂ I there is A ∈ Γ with E ⊂ A . Indeed, byTheorem 4 and Lemma 23 (applied to X = D ), there exists a countable set M such that E ⊂ M ∩ I and { x | I ∩ M : x ∈ D } = D ∩ M , which implies that M ∩ I ∈ Γ. Finally, note that the cofinality of Γ in [ I ] ≤ ω implies that thenet ( r A ) A ∈ Γ converges pointwise to the identity in K . Thus, Γ is cofinal and σ -complete in [ I ] ≤ ω and ( r A ) A ∈ Γ is a commutative retractional skeleton on K inducing the the set D . (cid:3) Proof of Lemma 22.
Let S and Φ be the union of the countable sets andfinite lists of formulas from the statements of Lemma 7, Lemma 8, Theo-rem 15 and Lemma 23. Pick M ≺ (Φ; S ∪ { K, D, I, τ } ). Since D is densein K and contained in the set induced by a retractional skeleton (see e.g.Lemma 24), by Theorem 15, M admits canonical retraction r M associatedto M , K and D .By Lemma 23, using the continuity of the mapping K ∋ x x | I ∩ M andcompactness of K , we have D ∩ M = { x | I ∩ M : x ∈ K } and so the retraction r is well-defined, continuous and r [ K ] = D ∩ M . Let π : I → R K be themapping given for i ∈ I and x ∈ K as π ( i )( x ) := x ( i ). By Lemma 8 (4)and (6), we have π ∈ M and π [ I ] ∩ M = π [ I ∩ M ]. Since we obviously have f ◦ r = f , for every f ∈ π [ I ∩ M ] = π [ I ] ∩ M and π [ I ] ∈ M separates thepoints of K , using Theorem 15 we obtain that r = r M . (cid:3) Valdivia embedding.
For compact spaces of weight ω admitting acommutative retractional skeleton, suitable models provide a very concretedescription of the “Valdivia embedding”. Theorem 25.
Let K be a compact space with w( K ) = ω , A ⊂ C ( K ) be a bounded set that separates the points of K and λ > be such that A ⊂ λB C ( K ) . Assume that K admits a retractional skeleton s = ( r s ) s ∈ Γ andlet ( M α ) α<ω be sets satisfying (Ra)-(Rd) and ( r α ) α<ω be the retractionssatisfying (R1)-(R10). For each α < ω , define T α = ( A ∩ M α ) × { α } andset T = T ∪ S α<ω T α +1 . If A ∈ M , then the mapping h : K → [ − , T defined as h ( x )( t ) := ( λ ( f ( x ) − f ( r α ( x ))) , t = ( f, α + 1) , f ∈ A ∩ M α +1 , λ f ( r ( x )) , t = ( f, , f ∈ A ∩ M . is a homeomorphic embedding and h (cid:2) D ( s )] ⊂ Σ( T ) . Moreover, ( r α ) α<ω iscommutative retractional skeleton inducing the set D ( s ) .Proof. Clearly h is continuous. Let us verify that it is one-to-one. Indeed,if x, y are distinct points from K then by (R7) there is a minimal ordinal α < ω for which r α ( x ) = r α ( y ) and α = 0 or it is a successor ordinal.If α = 0, then by (R10) there exists f ∈ M ∩ A such that f (cid:0) r ( x ) (cid:1) = f (cid:0) r ( y ) (cid:1) and so we have h ( x )( f, = h ( y )( f, α = β + 1for some β < ω , then by (R10) there exists f ∈ A ∩ M β +1 such that f ( x ) = f ( r β +1 ( x )) = f ( r β +1 ( y )) = f ( y ). Therefore, since r β ( x ) = r β ( y ),we obtain that h ( x )( f, β + 1) = h ( y )( f, β + 1). Note that it follows from(R1), (R2), (R3) and (R5) that ( r α ) α<ω is a retractional skeleton on K ,since (Rb) ensures that M α is countable, for every α < ω (where on ω we consider the ordering α ≤ ∗ β if and only if M α ⊂ M β ). Moreover byTheorem 15(ii)(c) and [7, Lemma 3.2] we have that S α<ω r α [ K ] = D ( s ).Now let us show that h [ D ( s )] ⊂ Σ( T ). If x ∈ D ( s ), then there exists aminimal α < ω such that r α ( x ) = x . We claim thatsuppt (cid:0) h ( x ) (cid:1) = { t ∈ T : h ( x )( t ) = 0 } ⊂ T ∪ [ α<α +1 T α +1 . Indeed, if t ∈ T with t / ∈ T ∪ S α<α +1 T α +1 , then there exist α > α and f ∈ M α +1 ∩ A such that t = ( f, α + 1). Thus, h ( x )( t ) = λ (cid:0) f ( x ) − f (cid:0) r α ( x ) (cid:1)(cid:1) = 0 , since α > α implies that r α ( x ) = x . This proves the claim. Finally thefact that h [ D ( s )] ⊂ Σ( T ), follows by observing that T is countable as wellas T α +1 , for every α < ω . (cid:3) On the other hand, for compact spaces of higher densities we do notknow a direct formula for the homeomorphic embedding. Still, an inductiveargument gives us Theorem C whose proof is based on the proof of [5,Theorem 2.6]. Quite surprisingly, the inductive argument does not give
HARACTERIZATION OF EBERLEIN COMPACTA 23 us only the “Valdivia embedding” (that is, Theorem C (i) ⇒ (iv)), but italso provides us with a new characterization of Valdivia compacta (that is,Theorem C (i) ⇔ (iii)) which we use later. Let us also highlight that there isan analogy of this new characterization in the language of suitable models,see Theorem 28. We start with a lemma. Lemma 26.
Let K be a compact space and let A ⊂ C ( K ) be a set separatingthe points of K . Then there exists A ′ ⊂ A with |A ′ | = w( K ) which separatesthe points of K .Proof. First, since by the Stone-Weierstrass theorem alg(
A ∪ { } ) is densein C ( K ), we easily observe that { f − ( − / , /
2) : f ∈ alg( A ∪ { } ) } is abasis for the topology of K . Thus, by [11, Theorem 1.1.15], there is F ⊂ alg(
A ∪ { } ) with |F | = w( K ) such that { f − ( − / , /
2) : f ∈ F } is abasis for the topology of K . Pick A ′ ⊂ A such that |A ′ | = w( K ) and F ⊂ alg( A ′ ∪ { } ). Then A ′ separates the points of K , because otherwisealg( A ′ ∪ { } ) would not separate the points of K , a contradiction with thefact that { f − ( − / , /
2) : f ∈ F } is a basis of the topology. (cid:3) Proof of Theorem C. (i) ⇒ (ii) Let s be the commutative retractional skele-ton on K inducing D ( s ). Then by Theorem 21 there is a weak subskeleton ofboth s and s , which easily implies that there is a cofinal subset Γ ′′ ⊂ Γ suchthat r s ◦ r t = r t ◦ r s , for every s, t ∈ Γ ′′ . Thus, it suffices to let Γ ′ = (Γ ′′ ) σ .(ii) ⇒ (iii): Let s = ( r s ) s ∈ Γ ′ be a commutative subskeleton of s . Pick anup-directed set Γ ′′ ⊂ Γ ′ and x ∈ D ( s ) = D ( s ). By Lemma 2 the limitlim s ∈ Γ ′′ r s ( x ) exists. Since there exists s ∈ Γ ′ such that x = r s x , using thecommutativity we obtainlim s ∈ Γ ′′ r s ( x ) = lim s ∈ Γ ′′ r s ( r s x ) = r s ( lim s ∈ Γ ′′ r s ( x )) ∈ D ( s ) . Thus, s satisfies (iii) with D = D ( s ).(iii) ⇒ (iv): First, we may without loss of generality assume that s = s . Wewill prove the result by induction on κ := w( K ). We may without loss ofgenerality assume that λ = 1. If κ = ω , then by Lemma 26, there exists acountable set H ⊂ A which separates the points of K and this set does thejob. So let us assume that the result holds for every compact space of weightstrictly smaller than κ . Proposition 17 together with Theorem 4 implythe existence of sets ( M α ) α ≤ κ satisfying (Ra)-(Rd) and retractions ( r α ) α ≤ κ satisfying (R1)-(R11). Note that using (R8), we obtain that r α [ D ] ⊂ D .For every α < κ , define A α := { f ∈ C ( r α [ K ]) : f ◦ r α ∈ A} . It is easy to seethat, for every α < κ , the set A α is symmetric, closed, convex and bounded.The fact that A α separates the points of r α [ K ] follows from (R6) and (R9)implies that f ◦ r s ∈ A α , for every f ∈ A α and every s ∈ (Γ ∩ M α ) σ . Forevery α < κ , define D α := D ∩ r α [ K ] ⊂ D ( s ) ∩ r α [ K ]. Since r α [ D ] is densein r α [ K ] and r α [ D ] ⊂ D α , we have that D α is dense in r α [ K ]. Therefore theinduction hypothesis and (R5) imply that there are sets T α ⊂ A α such thatthe mapping ϕ α : r α [ K ] → [ − , T α given by ϕ α ( x )( t ) := t ( x ), t ∈ T α and x ∈ r α [ K ] is a homeomorphic embedding and ϕ α [ D α ] ⊂ Σ( T α ), for every α < κ . We may without loss of generality assume that T α ∩ T β = ∅ for α = β . Now, we put T = T ∪ S α<κ T α +1 and define ϕ : K → [ − , T by ϕ ( x )( t ) := ( (cid:0) ϕ α +1 ( r α +1 ( x ))( t ) − ϕ α +1 ( r α ( x ))( t ) (cid:1) , t ∈ T α +1 ,ϕ ( r ( x ))( t ) , t ∈ T . Then ϕ is of course continuous. Let us verify that it is one-to-one. Indeed,if x, y are distinct points from K then by (R7) there is a minimal ordinal α < κ for which r α ( x ) = r α ( y ) and α = 0 or it is a successor ordinal. If α = 0, then there exists t ∈ T such that ϕ (cid:0) r ( x ) (cid:1) ( t ) = ϕ (cid:0) r ( y ) (cid:1) ( t ) andso we have ϕ ( x )( t ) = ϕ ( y )( t ). Otherwise, α = β + 1 for some β < κ andthere is t ∈ T α such that ϕ α ( r α ( x ))( t ) = ϕ α ( r α ( y ))( t ). Moreover, since α is minimal, we have r β ( x ) = r β ( y ), hence we obtain ϕ ( x )( t ) = ϕ ( y )( t )and so ϕ ( x ) = ϕ ( y ). Thus, ϕ is a homeomorphic embedding.Let us show that ϕ [ D ( s )] ⊂ Σ( T ). Indeed, by (R11), for every x ∈ D theset { α < κ : r α +1 ( x ) = r α ( x ) } is countable. Moreover, since r α [ D ] ⊂ D α , theinduction hypothesis ensures that the supports of ϕ (cid:0) r ( x ) (cid:1) , ϕ α +1 ( r α +1 ( x ))and ϕ α +1 ( r α ( x )) are countable. Therefore the support of ϕ ( x ) is countableand we obtain ϕ [ D ] ⊂ Σ( T ). Moreover, since D is dense in K , by Lemma 24there is a commutative retractional skeleton s on ϕ [ K ] such that D ( s ) = ϕ [ K ] ∩ Σ( T ). Since ϕ [ D ( s )] ∩ D ( s ) ⊃ ϕ [ D ], by [7, Lemma 3.2] we have that ϕ [ D ( s )] = D ( s ) ⊂ Σ( T ).Now, let us show that π t ◦ ϕ ∈ A , for every t ∈ T . Firstly, note that π t ◦ ϕ α ∈ A α , for every t ∈ T α and every α < κ . If t ∈ T , then we havethat π t ◦ ϕ = π t ◦ ϕ ◦ r ∈ A . Pick α < κ and t ∈ T α +1 . Then, similarly asabove, π t ◦ ϕ α +1 ◦ r α +1 ∈ A and therefore using (R6), we obtain π t ◦ ϕ = (cid:0) π t ◦ ϕ α +1 ◦ r α +1 − π t ◦ ϕ α +1 ◦ r α +1 ◦ r α (cid:1) ∈ A . Omitting some indices, we may without loss of generality assume that themapping T ∋ t f t := π t ◦ ϕ ∈ A is one-to-one and so H := { f t : t ∈ T } does the job.(iv) ⇒ (i): By Lemma 24 there is a commutative retractional skeleton s on ϕ [ K ] such that D ( s ) = ϕ [ K ] ∩ Σ( I ). Since ϕ [ D ( s )] ⊂ D ( s ), by [7, Lemma3.2] we have that ϕ [ D ( s )] = D ( s ) and so the set D ( s ) is induced by acommutative retractional skeleton.(iii) ⇒ (i): follows from (iii) ⇒ (iv) ⇒ (i) applied to the set A := B C ( K ) . (cid:3) The following corollary might be well-known, but let us mention it forfuture reference.
Corollary 27.
Let K be a compact space and let s be a full retractionalskeleton on K . Then there exists a commutative subskeleton of s .Proof. Apply Theorem C (iii) = ⇒ (ii) to D := K . (cid:3) The proof of Theorem C gives us also the following.
HARACTERIZATION OF EBERLEIN COMPACTA 25
Theorem 28.
Let K be a compact space and let D be a set induced by aretractional skeleton on K . Then the following are equivalent.(a) D is induced by a commutative retractional skeleton.(b) For every suitable model M the following holds: ∀ x ∈ D ∃ y ∈ D ∩ D ∩ M ∀ f ∈ C ( K ) ∩ M : f ( x ) = f ( y ) . Proof. (a) ⇒ (b): By Theorem 15, there is a finite list of formulas Φ and acountable set S (depending on the compact space K and the set D ) suchthat for any M ≺ (Φ , S ), M admits canonical retraction r M and we have r M [ D ] ⊂ D . Then (b) follows from Theorem 15 (i) applied to A := C ( K ).(b) ⇒ (a): Follows from the fact that in the proof of Theorem C (iii) ⇒ (iv)we used condition (iii) only to ensure that for a suitable model M α wehave r α [ D ] ⊂ D , which by Theorem 15 (i) follows from the condition (b)above. (cid:3) Characterization of (semi-)Eberlein compacta
Here, we apply the results of the preceding sections and characterize(semi)-Eberlein compacta using the notion of an A -shrinking retractionalskeleton.5.1. Main results and their consequences.
Let us start with the defi-nition of an A -shrinking retractional skeleton. Definition 29.
Let K be a countably compact space. Let ∅ 6 = A ⊂ C ( K )be a bounded set. The pseudometric ρ A on K is given as ρ A ( k, l ) := sup f ∈A | f ( k ) − f ( l ) | , k, l ∈ K. If ( r s ) s ∈ Γ is a retractional skeleton on K and D ⊂ K , we say that ( r s ) s ∈ Γ is A -shrinking with respect to D if for every x ∈ D and every increasing se-quence ( s n ) n ∈ ω in Γ with s := sup n ∈ ω s n , we have that lim n ∈ ω ρ A ( r s n ( x ) , r s ( x )) =0. If ( r s ) s ∈ Γ is A -shrinking with respect to K , then we just write that ( r s ) s ∈ Γ is A -shrinking .Note that if the nonempty and bounded set A separates the points of K ,then ρ A is a metric on K .The aim of this section is to prove the following result from which Theo-rem A and Theorem B follow. Theorem 30.
Let K be a compact space and let D ⊂ K be a dense set.Consider the following conditions.(i) There exists a homeomorphic embedding h : K → [ − , I such that h [ D ] = c ( I ) ∩ h [ K ] .(ii) There exists a bounded set A ⊂ C ( K ) separating the points of K anda retractional skeleton s = ( r s ) s ∈ Γ on K with D ⊂ D ( s ) such that(a) s is A -shrinking with respect to D ,(b) f ◦ r s ∈ A , for every f ∈ A and s ∈ Γ , and (c) lim s ∈ Γ ′ r s ( x ) ∈ D , for every x ∈ D and every up-directed subset Γ ′ of Γ .(iii) There exists a homeomorphic embedding h : K → [ − , J such that h [ D ] ⊂ c ( J ) .Then (i) ⇒ (ii) ⇒ (iii). For compact spaces of weight ω we have even a better result. The mostimportant difference here is that we do not need conditions (b) and (c) fromTheorem 30. Theorem 31.
Let K be a compact space with w( K ) = ω . Assume thatthere exist a bounded set A ⊂ C ( K ) that separates the points of K and aretractional skeleton s = ( r s ) s ∈ Γ on K that is A -shrinking with respect to D ,where D ⊂ D ( s ) . Then there exists a homeomorphic embedding Φ : K → [ − , T such Φ (cid:2) D ( s ) (cid:3) ⊂ Σ( T ) and Φ[ D ] ⊂ c ( T ) . Corollary 32.
Let K be a compact space with w( K ) = ω . Then K isEberlein (resp. semi-Eberlein) if and only if there exist a bounded set A ⊂C ( K ) which separates the points of K and a retractional skeleton s on K which is A -shrinking (resp. A -shrinking with respect to a dense subset of D ( s ) ).Proof. The “only if” part follows immediately from (the easier part of) The-orem A and Theorem B which give even stronger conditions, the “if” partis an immediate consequence of Theorem 31 and the fact that if s is A -shrinking, then s is full (see Lemma 35). (cid:3) Let us first comment on those results and their consequences.
Remark . The notion of a shrinking retractional skeleton is inspired by[14], where the definition of a shrinking projectional skeleton was given andWCG Banach spaces were characterized using this notion.Given a retractional skeleton ( r s ) s ∈ Γ on a compact space K , it is wellknown that ( P s ) s ∈ Γ given by P s ( f ) := f ◦ r s , s ∈ Γ is a projectional skeletonon C ( K ), see e.g. [20, Proposition 5.3]. Moreover, if ∅ 6 = A ⊂ C ( K ) is abounded set and ( P s ) s ∈ Γ is A -shrinking in the sense of [14, Definition 16],it is not very difficult to observe that ( r s ) s ∈ Γ is A -shrinking in the sense ofDefinition 29. It is not clear whether the converse holds as it is (at leastformally) a stronger condition. Thus, Theorem A allows us in a certainsense to strengthen implication (ii) ⇒ (i) from [14, Theorem 21]. Since theother implication is easier, Theorem A may be thought of as a topologicalcounterpart and in a certain sense also strengthening of [14, Theorem 21] inthe context of C ( K ) spaces. Proof of Theorem B.
Assume that K is semi-Eberlein and fix a homeomor-phic embedding h : K → [ − , I such that h [ K ] ∩ c ( I ) is dense in h [ K ].Defining D := h − [ c ( I )], the assertion follows from Theorem 30 (i) ⇒ (ii).The other implication follows from Theorem 30 (ii) ⇒ (iii). (cid:3) HARACTERIZATION OF EBERLEIN COMPACTA 27
Instead of Theorem A we shall prove a slightly more general result whereinstead of compactness we assume countable compactness, see Theorem 38.In order to show the argument, we need three lemmas first.The first one is [14, Proposition 20], we recall it here for further references.
Lemma 34.
Let ( X, ρ ) be a metric space, Γ be an up-directed and σ -complete partially ordered set, Γ ′ be an up-directed subset of Γ and ( x γ ) γ ∈ Γ be a net in X . If for every increasing sequence ( γ n ) n ≥ in Γ ′ with γ =sup n ≥ γ n ∈ Γ , it holds that lim n →∞ ρ ( x γ n , x γ ) = 0 , then the net ( x γ ) γ ∈ Γ ′ is convergent. More precisely, there exists an increasing sequence ( t n ) n ≥ in Γ ′ with t = sup n ≥ t n such that lim γ ∈ Γ ′ x γ = x t . The second one is a generalization of the fact that any A -shrinking re-tractional skeleton is also full, whenever A ⊂ C ( K ) separates the points of K . Lemma 35.
Let K be a compact space, A ⊂ C ( K ) be a bounded set separat-ing the point of K and let s = ( r s ) s ∈ Γ be a retractional skeleton on K whichis A -shrinking with respect to a set D with D ⊃ D ( s ) . Then D = D ( s ) and lim s ∈ Γ ′ r s ( x ) ∈ D , for every x ∈ D and every up-directed subset Γ ′ of Γ .Proof. Fix x ∈ D and an up-directed set Γ ′ ⊂ Γ. Since s is A -shrinking withrespect to D , by Lemma 34 there exists an increasing sequence ( s n ) n ∈ ω in Γ ′ with s = sup n s n ∈ Γ such that ρ A − lim t ∈ Γ ′ r t ( x ) = r s ( x ). Therefore, sincethe limit lim t ∈ Γ ′ r t ( x ) exists, we obtain f (lim t ∈ Γ ′ r t ( x )) = f ( r s ( x )), for every f ∈ A . Since A separates the points of K , we deduce that lim t ∈ Γ ′ r t ( x ) = r s ( x ) ∈ D ( s ) ⊂ D . Finally, for Γ ′ = Γ we obtain x = lim s ∈ Γ r s ( x ) ∈ D ( s )and so D ⊂ D ( s ). (cid:3) For the third lemma recall that every real-valued continuous functiondefined on a countably compact space D is bounded so we may consider thesupremum norm on C ( D ). Lemma 36.
Let D be a countably compact space. Suppose that there exista bounded set A ⊂ C ( D ) separating the points of D and a full retractionalskeleton s = ( r s ) s ∈ Γ on D such that f ◦ r s ∈ A , for every f ∈ A and s ∈ Γ .Then A ′ = { βf : f ∈ A} separates the points of βD .Proof. By [10, Proposition 4.5], there exists a retractional skeleton S =( R s ) s ∈ Γ on βD such that D ( S ) = D and R s | D = r s , for every s ∈ Γ. Let x, y ∈ βD be distinct points. Since lim s ∈ Γ R s ( x ) = x and lim s ∈ Γ R s ( y ) = y ,there exists s ∈ Γ such that R s ( x ) = R s ( y ). Since R s ( x ) , R s ( y ) ∈ D , thereexists a function f ∈ A such that f ( R s ( x )) = f ( R s ( y )). Therefore we have βf ( R s ( x )) = βf ( R s ( y )). It is easy to see that ( βf ◦ R s ) | D = f ◦ r s , whichimplies that βf ◦ R s = β ( f ◦ r s ) ∈ A ′ , since f ◦ r s ∈ A . (cid:3) Remark . Note that the assumption “ f ◦ r s ∈ A , for every f ∈ A and s ∈ Γ” in Lemma 36 is essential. Indeed, consider D = [0 , ω ) and A = { { }∪ [ α +1 ,ω ) : α < ω } . Then it is easy to see that A separates the points of D and that D admitsthe full retractional skeleton ( r α ) α<ω given by the formula r α ( β ) = ( β β ≤ αα + 1 α < β < ω , for every α < ω . However, we have βD = [0 , ω ] and the set A ′ = { β { }∪ [ α +1 ,ω ) : α < ω } = { { }∪ [ α +1 ,ω ] : α < ω } does not separate 0 from ω . Theorem 38.
Let D be a countably compact space. Then the followingconditions are equivalent.(i) There exists a set I such that D embeds homeomorphically into ( c ( I ) , w ) .(ii) D is an Eberlein compact space.(iii) There exist a bounded set A ⊂ C ( D ) separating the points of D anda full retractional skeleton s = ( r s ) s ∈ Γ on D such that(a) s is A -shrinking,(b) f ◦ r s ∈ A , for every f ∈ A and s ∈ Γ .Proof. ( i ) ⇒ ( ii ) follows from the classical Eberlein-ˇSmulian theorem and( ii ) ⇒ ( iii ) follows from Theorem 30 and Lemma 35. If (iii) holds, pickthe corresponding set A and the full retractional skeleton ( r s ) s ∈ Γ on D . By[10, Proposition 4.5], there exists a retractional skeleton S = ( R s ) s ∈ Γ on βD such that D ( S ) = D and R s | D = r s , for every s ∈ Γ. Consider nowthe set A ′ := { βf : f ∈ A} ⊂ C ( βD ) and the retractional skeleton S . ByLemma 36, A ′ separates the points of βD . Obviously, S is A ′ -shrinking withrespect to D and it is easy to see that for every f ∈ A and s ∈ Γ we have βf ◦ R s = β ( f ◦ r s ) ∈ A ′ . By Lemma 35 we have that lim s ∈ Γ ′ R s ( x ) ∈ D , forevery x ∈ D and every up-directed subset Γ ′ of Γ. Therefore, Theorem 30ensures that (i) holds. (cid:3) Finally, let us mention that the shrinkingness of a retractional skeleton isnot a specific property of one particular skeleton.
Lemma 39.
Let K be a compact space and A ⊂ C ( K ) be a bounded set sep-arating the points of K . If there exists an A -shrinking retractional skeletonon K , then every full retractional skeleton on K admits a weak subskeletonwhich is A -shrinking and commutative.Proof. By Lemma 35, there exists an A -shrinking and full retractional skele-ton ( e r i ) i ∈ I on K . Moreover, by Corollary 27 we may without loss of general-ity assume that ( e r i ) i ∈ I is commutative. Now, let ( r s ) s ∈ Γ be a full retractionalskeleton on K . By Theorem 21, there exists a retractional skeleton ( R i ) i ∈ Λ which is a weak subskeleton of both ( r s ) s ∈ Γ and ( e r i ) i ∈ I . It is easy to seethat ( R i ) i ∈ Λ is commutative and A -shrinking. (cid:3) HARACTERIZATION OF EBERLEIN COMPACTA 29
Proofs of Theorem 30 and Theorem 31.
Let us start with theproof of Theorem 30 (i) ⇒ (ii). Lemma 40.
Let K be a compact space and D ⊂ K be a dense subset. Ifthere exists a homeomorphic embedding h : K → [ − , I such that h [ D ] = h [ K ] ∩ c ( I ) , then (ii) in Theorem 30 holds with the same set D .Proof. We may without loss of generality assume that K ⊂ [ − , I , D = K ∩ c ( I ). Pick the commutative retractional skeleton ( r A ) A ∈ Γ from Lemma 24and put S := { π i | K : i ∈ I } ∪ { } ⊂ C ( K ). Clearly S is bounded andseparating. Moreover D is obviously contained in the set K ∩ Σ( I ) (whichis the set induced by ( r A ) A ∈ Γ ) and it is easy to observe that if A ∈ Γ and f ∈ S , then f ◦ r A ∈ S .Now, let us show that ( r A ) A ∈ Γ is S -shrinking with respect to D . Pick x ∈ D , an increasing sequence ( A n ) n ∈ ω in Γ and put A = sup n A n = S n ∈ ω A n .Fix ε > n ∈ ω be such that { i ∈ A : | x ( i ) | > ε } ⊂ A n . Then forevery n ≥ n we obtain r A n x ( i ) − r A x ( i ) = 1 A \ A n · x ( i ), therefore for every i ∈ I we have | r A n x ( i ) − r A x ( i ) | < ε ; hence sup f ∈S | f ( r A n x ) − f ( r A x ) | < ε .Finally, we note that whenever Γ ′ ⊂ Γ is up-directed and x ∈ D , then y := lim A ∈ Γ ′ r A x exists by Lemma 2 and moreover if i ∈ suppt y , then y ( i ) = x ( i ). Therefore, we have that y ∈ K ∩ c ( I ) = D . (cid:3) Now we present the proof of Theorem 31. This proof is based on our veryconcrete knowledge of the “Valdivia embedding” given by Theorem 25.
Proof of Theorem 31.
It follows from Theorem 4 and Proposition 17 thatthere exist sets ( M α ) α<ω satisfying (Ra)-(Rd) such that A ∈ M and re-tractions ( r α ) α<ω satisfying (R1)-(R10). Therefore if λ > A ⊂ λB C ( K ) and h : K → [ − , T is the mapping defined in the statementof Theorem 25, then h is a homeomorphic embedding, h (cid:2) D ( s ) (cid:3) ⊂ Σ( T )and ( r α ) α<ω is a commutative retractional skeleton inducing the set D ( s ).Since s is A -shrinking with respect to D , using that the mapping ω ∋ α sup( M α ∩ Γ) ∈ Γ is increasing and Theorem 15(ii)(c), we get that ( r α ) α<ω is A -shrinking with respect to D .Now consider the mapping Φ : K → [ − , T given byΦ( x )( f α +1 n , α +1) := 1 n h ( x )( f α +1 n , α +1) and Φ( x )( f n ,
0) := 1 n h ( x )( f n , , where A ∩ M = ( f n ) n< |A∩ M | and A ∩ M α +1 = ( f α +1 n ) n< |A∩ M α +1 | , forevery α < ω . It is clear that Φ is a homeomorphic embedding and thatΦ[ D ( s )] ⊂ Σ( T ). We claim that Φ[ D ] ⊂ c ( T ). Fix x ∈ D and note thatto prove that Φ( x ) ∈ c ( T ), it is enough to show that, for every ε >
0, thefollowing set is finiteΛ := { α < ω : | h ( x )( t ) | > ε for some t ∈ T α +1 } . Suppose by contradiction that Λ is infinite. Take a strictly increasing se-quence ( α k ) k ∈ ω from Λ and functions f k ∈ A∩ M α k +1 such that | h ( x )( f k , α k + | > ε . Let α = sup k ∈ ω α k and fix k ∈ ω . Then ρ A ( r α k ( x ) , r α ( x )) = sup f ∈A |h r α k ( x ) − r α ( x ) , f i| ≥ |h r α k ( x ) − r α ( x ) , f k i| = | f k ( r α ( x )) − f k ( r α k ( x )) | . Observing that M α k +1 ⊂ M α , (R10) ensures that f k ( r α ( x )) = f k ( x ). There-fore we obtain ρ A ( r α k ( x ) , r α ( x )) ≥ | f k ( x ) − f k ( r α k ( x )) | > λε. But this contradicts the fact that the skeleton ( r α ) α<ω is A -shrinking withrespect to D . (cid:3) For spaces of weight greater than ω the situation is more complicated,because our knowledge of the “Valdivia embedding” is somehow limited, seeTheorem C. Let us start with an easy observation. Lemma 41.
Let K be a compact space, A ⊂ C ( K ) be a bounded set sep-arating the points of K and D be a subset of K . Suppose that ( r s ) s ∈ Γ isa retractional skeleton on K that is A -shrinking with respect to D . For anup-directed set Γ ′ ⊂ Γ let R Γ ′ be as in Lemma 2. Then for every x ∈ D wehave the following.(1) If Γ ′ ⊂ Γ is up-directed, then R Γ ′ ( x ) = ρ A - lim s ∈ Γ ′ r s ( x ) .(2) If M ⊂ P (Γ) is such that M is up-directed and each M ∈ M isup-directed, then ρ A - lim M ∈M R M ( x ) = R S M ( x ) .Proof. Pick x ∈ D .(1) Applying Lemma 34, we see that the limit ρ A - lim s ∈ Γ ′ r s ( x ) exists, let usdenote it by y ∈ K . Moreover, for every f ∈ A we have that f ( R Γ ′ ( x )) = f ( y ). Since A separates the points of K , we obtain that y = R Γ ′ ( x ).(2) Pick ε >
0. By (1), there exists s ∈ S M such that ρ A ( r s ( x ) , R S M ( x )) <ε , for every s ≥ s . Let M ∈ M be such that s ∈ M . Then, for every M ∈ M with M ⊃ M by (1), we have ρ A ( R M ( x ) , R S M ( x )) = lim s ∈ M,s ≥ s ρ A ( r s ( x ) , R S M ( x )) ≤ ε. (cid:3) The following proposition together with Theorem C is the core of ourargument. The idea to use such a result is related to a characterizationof Eberlein compacta by Farmaki [15, Theorem 2.9] (see also [12, Theorem10]). Note however, that our methods enable us to present a self-containedproof.
Proposition 42.
Let K ⊂ [ − , I be a compact space and define S K = { π i | K : i ∈ I } . Suppose that K admits a retractional skeleton s = ( r s ) s ∈ Γ such that D ( s ) ⊂ Σ( I ) and let D ⊂ D ( s ) . If s is S K -shrinking with respect to D and lim s ∈ Γ ′ r s ( x ) ∈ HARACTERIZATION OF EBERLEIN COMPACTA 31 D , for every x ∈ D and every up-directed subset Γ ′ of Γ , then for every ε > there is a decomposition I = S ∞ n =0 I εn such that ∀ n ∀ x ∈ D : |{ i ∈ I εn : | x ( i ) | > ε }| < ω. Proof.
By [18, Proposition 19.5], we may pick a set J ⊂ I such that | J | =w( K ) and suppt x ⊂ J , for every x ∈ K . By [7, Lemma 3.2], we havethat D ( s ) = Σ( I ) ∩ K and hence Lemma 24 ensures that D ( s ) is inducedby a commutative retractional skeleton. Therefore it follows from TheoremC that we may assume that the retractional skeleton s is commutative.Now, let us prove the result by induction on the weight of K . If K hascountable weight, then the set J is countable and we may enumerates it as J := ( j n ) n ≥ . For each ε >
0, let I ε = I \ J , and I εn = { j n } , for every n ≥ I = S ∞ n =0 I εn and ∀ n ∀ x ∈ K : |{ i ∈ I εn : | x ( i ) | > ε }| ≤ . Now suppose that w( K ) = κ ≥ ω and that the result holds for compactspaces of weight less than κ . Proposition 17 together with Theorem 4 implythe existence of sets ( M α ) α ≤ κ satisfying (Ra)-(Rd) and retractions ( r α ) α ≤ κ satisfying (R1)-(R11). Note that we can assume that J ⊂ S α<κ M α , byreplacing (Rc) by the following (stronger) condition: M α +1 ≺ (Φ; { j α , α } ∪ M α ) , ∀ α < κ, where J = { j α : α < κ } . Note that, by Lemma 22, we may assume that r α ( x ) = x | I ∩ M α , for every x ∈ K and α < κ . For each α < κ , it is easy tosee that the retractional skeleton ( r s | r α [ K ] ) s ∈ (Γ ∩ M α ) σ given by (R5) is S r α [ K ] -shrinking with respect to the set D ∩ r α [ K ] ⊂ D ( s ) ∩ r α [ K ] ⊂ Σ( I ) ∩ [ − , I .Moreover, if Γ ′ ⊂ (Γ ∩ M α ) σ is up-directed and x ∈ D ∩ r α [ K ], then using(R9) we conclude that lim s ∈ Γ ′ r s ( x ) ∈ D ∩ r α [ K ]. Now fix ε > I = S n ≥ I εn, be the decomposition given by induction hypothesis appliedto r [ K ] (using that by (R5) we may apply the inductive hypothesis to r [ K ]), that is, for every y ∈ D ∩ r [ K ] and n ≥ { i ∈ I εn, : | y ( i ) | > ε } is finite. Fix α < κ , similarly let I = S n ≥ I εn,α +1 be the decompositiongiven by the induction hypothesis applied to r α +1 [ K ], that is, for every y ∈ D ∩ r α +1 [ K ] and n ≥ { i ∈ I εn,α +1 : | y ( i ) | > ε } is finite. Define I ε = I \ J and for every n ≥
1, put I εn = ( J ∩ I εn, ∩ M ) ∪ [ α<κ (cid:0) J ∩ I εn,α +1 ∩ ( M α +1 \ M α ) (cid:1) . Note that I εn ∩ I εm = ∅ , if n = m and that I = S n ≥ I εn , since J ⊂ S α<κ M α .Fixed x ∈ D and n ≥
0, let us show that the set S n = { i ∈ I εn : | x ( i ) | > ε } is finite. Since suppt( x ) ⊂ J , we have that S is empty. Fixed n ≥
1, notethat in order to conclude that S n is finite it suffices to prove that the setΛ = { α < κ : | x ( i ) | > ε for some i ∈ J ∩ I εn,α +1 ∩ ( M α +1 \ M α ) } is finite. Indeed, using that r ( x ) = x | I ∩ M = x | J ∩ M we obtain that: S n ∩ ( J ∩ I εn, ∩ M ) = { i ∈ I εn, : | r ( x )( i ) | > ε } and therefore, since r ( x ) = lim s ∈ (Γ ∩ M ) r s ( x ) ∈ D ∩ r [ K ], we conclude that S n ∩ ( J ∩ I εn, ∩ M ) is finite. Similarly, for α < κ we have S n ∩ ( J ∩ I εn,α +1 ∩ M α +1 \ M α ) = { i ∈ I εn,α +1 : | r α +1 ( x ) | I \ M α ( i ) | > ε }⊂ { i ∈ I εn,α +1 : | r α +1 ( x )( i ) | > ε } and therefore S n ∩ ( J ∩ I εn,α +1 ∩ M α +1 \ M α ) is finite. It remains to provethat Λ is finite. In order to do that suppose by contradiction that Λ isinfinite, so there is a strictly increasing sequence ( α k ) k ≥ of elements of κ and a sequence ( i k ) k ≥ such that i k ∈ J ∩ I εn,α k +1 ∩ ( M α k +1 \ M α k ) and | x ( i k ) | > ε , for every k ≥
1. Put α = sup k α k . Then we have (because i k ∈ M α k +1 \ M α k ⊂ M α \ M α k ): ε < | x ( i k ) | = | r α ( x )( i k ) − r α k ( x )( i k ) | ≤ ρ S K ( r α ( x ) , r α k ( x )) , for every k ≥
1. This is a contradiction, because using (R8) and Lemma 41(2) applied to M = { M α k ∩ Γ : k ≥ } , we conclude that lim k →∞ ρ S K ( r α ( x ) , r α k ( x )) =0, since s is S K -shrinking with respect to D . (cid:3) The following is based on [12, Theorem 10].
Proposition 43.
Let K ⊂ [ − , I be a compact space and D be a subset of K . If for every ε > , there exists a decomposition I = S n ∈ ω I εn such thatfor every x ∈ D and every n ∈ ω the set { i ∈ I εn : | x ( i ) | > ε } is finite, then there is a homeomorphic embedding Φ : K → [ − , I × ω suchthat Φ[ D ] ⊂ c ( I × ω ) .Proof. Let k ∈ ω and define the function τ k : R → R as τ k ( t ) = t + k , if t ≤ − k , , if − k ≤ t ≤ k ,t − k , if t ≥ k . Define then Φ : K → [ − , I × ω byΦ( x )( i, k ) = 1 nk τ k ( x ( i )) , if i ∈ I /kn , n ∈ ω and k ∈ ω . Since the map π ( i,k ) ◦ Φ : K → R is continuous,for every ( i, k ) ∈ I × ω , the map Φ is continuous as well. The map isalso one-to-one. Indeed, for distinct x , x ∈ K there exists an i ∈ I with x ( i ) = x ( i ). Let k ∈ ω be such that 1 /k < max {| x ( i ) | , | x ( i ) |} and pick HARACTERIZATION OF EBERLEIN COMPACTA 33 n ∈ ω with i ∈ I /kn . Then τ k ( x ( i )) = τ k ( x ( i )), therefore Φ( x )( i, k ) =Φ( x )( i, k ).It remains to prove that Φ[ D ] is contained in c ( I × ω ). In order to dothat, let x ∈ D and fix ε >
0. If n, k ∈ ω , and n > /ε or k > /ε , then | Φ( x )( i, k ) | < ε , for any choice of i ∈ I /kn . Let n, k < /ε (observe thatthere are only finitely many n and k such that this inequality holds). Thenwe have { i ∈ I /kn : | Φ( x )( i, k ) | > ε } ⊆ { i ∈ I /kn : τ k ( x ( i )) = 0 } = { i ∈ I /kn : x ( i ) > /k } ∪ { i ∈ I /kn : x ( i ) < − /k } = { i ∈ I /kn : | x ( i ) | > /k } . Therefore, the set { i ∈ I /kn : | Φ( x )( i, k ) | > ε } is finite and thus we concludethat Φ( x ) ∈ c ( I × ω ). (cid:3) Proof of Theorem 30.
Lemma 40 ensures that ( i ) ⇒ ( ii ). Now let us provethat (ii) ⇒ (iii). Let A and s = ( r s ) s ∈ Γ be as in the assumption. It is easy tosee that A ′ := conv( A ∪ −A ) satisfies the conditions (a) and (b) from (ii), sowe may without loss of generality assume that the set A is closed, convex andsymmetric. Let λ ≥ A ⊂ λB C ( K ) . By Theorem C, there exists H ⊂ A such that the mapping ϕ : K → [ − , H given by ϕ ( x )( h ) := λ h ( x ),for h ∈ H and x ∈ K , is a homeomorphic embedding and ϕ [ D ( s )] ⊂ Σ( H ).For every s ∈ Γ, define q s = ϕ ◦ r s ◦ ϕ − : ϕ [ K ] → ϕ [ K ] and note thatthe retractional skeleton ( q s ) s ∈ Γ is S ϕ [ K ] -shrinking with respect to the set ϕ [ D ] ⊂ ϕ [ D ( s )]. Indeed, fix x ∈ D and an increasing sequence ( s n ) n ≥ ofelements of Γ with s = sup n s n . Then we have0 ≤ ρ S ϕ [ K ] (cid:16) q s n (cid:0) ϕ ( x ) (cid:1) , q s (cid:0) ϕ ( x ) (cid:1)(cid:17) = sup h ∈H (cid:12)(cid:12)(cid:12) ϕ (cid:0) r s n ( x ) (cid:1) ( h ) − ϕ (cid:0) r s ( x ) (cid:1) ( h ) (cid:12)(cid:12)(cid:12) == 1 λ sup h ∈H (cid:12)(cid:12)(cid:12) h (cid:0) r s n ( x ) (cid:1) − h (cid:0) r s ( x ) (cid:1)(cid:12)(cid:12)(cid:12) ≤ λ sup f ∈A (cid:12)(cid:12)(cid:12) f (cid:0) r s n ( x ) (cid:1) − f (cid:0) r s ( x ) (cid:1)(cid:12)(cid:12)(cid:12) → . Moreover, for every up-directed set Γ ′ ⊂ Γ and every x ∈ D we have thatlim s ∈ Γ ′ q s ( ϕ ( x )) = ϕ (lim s ∈ Γ ′ r s ( x )) ∈ ϕ [ D ]. Therefore, the result followsfrom Proposition 42 and Proposition 43. (cid:3) Open questions and remarks
In Section 3 we obtained as an application of our methods that for acountable family of retractional skeletons inducing the same set there is acommon weak subskeleton, see Theorem 21. It would be interesting to knowwhether we can find even a subskeleton (not only a weak one).
Question 44.
In Theorem 21, is it possible to obtain a subskeleton insteadof a weak subskeleton?
When working with retractional skeletons, their index sets are quite mys-terious. For Banach spaces with a projectional skeleton, the index set maybe chosen to consist of the ranges of the involved projections (ordered by in-clusion), see [9, Theorem 4.1]. We wonder whether something similiar holdsfor spaces with a retractional skeleton.
Question 45.
Let K be a compact space and let D ⊂ K be induced by aretractional skeleton on K . Does there exist a family of retractions { r F : F ∈F } indexed by a family of compact spaces F satisfying the following condi-tions?(i) whenever ( F n ) n ∈ ω is an increasing sequence from F , then sup n F n = S n F n ∈ F ,(ii) for every F ∈ F we have r F [ K ] = F ,(iii) ( r F ) F ∈F is a retractional skeleton on K inducing the set D . Note that for compact spaces of weight ω we have a better character-ization of (semi)-Eberlein compacta than in the general case. We wonderwhether there is a way of improving it. Question 46.
Can we drop the assumption (b) (resp. (b) and (c)) from thecondition (2) in Theorem A (resp. Theorem B)?
Note that in Proposition 42 we proved a result in a sense very similar tothe characterization of Eberlein compacta from [15, Theorem 2.9] (see also[12, Theorem 10]). We wonder whether an analogoue of [12, Theorem 10]holds also in the context semi-Eberlein compacta. Note that one implicationfollows from Proposition 43, so a positive answer to the following questionwould give a characterization of semi-Eberlein compact subspaces of [ − , I . Question 47.
Let K ⊂ [ − , I be a compact space such that Σ( I ) ∩ K is dense in K . Let K be semi-Eberlein. Does there exist D ⊂ Σ( I ) ∩ K which is dense in K such that for every ε > there exists a decomposition I = S ∞ n =0 I εn satisfying ∀ n ∈ ω ∀ x ∈ D : |{ i ∈ I εn : | x ( i ) | > ε }| < ω ?Finally, let us emphasize that we believe that Theorem C gives quite abig flexibility to consider other subclasses of Valdivia compact spaces andcharacterize them using the notion of retractional skeletons. The reasonwhy we believe so is, that by Theorem C we may consider any set inducedby a retractional skeleton to be a subset of Σ( I ) (where A ⊂ C ( K ) playsthe role of the set I ); moreover, for subsets of Σ( I ) several classes of com-pact spaces were characterized using their evaluations on the set I , see [12].Thus, there is enough room for further possible research by considering thoseclasses of compacta and try to develop the right notion which would give acharacterization using retractional skeletons. HARACTERIZATION OF EBERLEIN COMPACTA 35
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Centro de Matem´atica, Computac¸˜ao e Cognic¸˜ao, Universidade Federal doABC, Avenida dos Estados, 5001, Santo Andr´e, Brasil
E-mail address : [email protected], [email protected] Faculty of Mathematics and Physics, Department of Mathematical Analy-sis, Charles University, 186 75 Praha 8, Czech Republic
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