An Asplund space with norming Markuševič basis that is not weakly compactly generated
Petr Hájek, Tommaso Russo, Jacopo Somaglia, Stevo Todorčević
aa r X i v : . [ m a t h . F A ] J u l AN ASPLUND SPACE WITH NORMING MARKUŠEVIČ BASISTHAT IS NOT WEAKLY COMPACTLY GENERATED
PETR HÁJEK, TOMMASO RUSSO, JACOPO SOMAGLIA, AND STEVO TODORČEVIĆ
Abstract.
We construct an Asplund Banach space X with a norming Markuševič basissuch that X is not weakly compactly generated. This solves a long-standing open problemfrom the early nineties, originally due to Gilles Godefroy. En route to the proof, weconstruct a peculiar example of scattered compact space, that also solves a question dueto Wiesław Kubiś and Arkady Leiderman. Introduction
The crystallisation, in the mid-sixties, of the notions of projectional resolution of theidentity (PRI, for short) [55] and of weakly compactly generated Banach space (WCG)[5] opened the way to a spectacular development in Banach space theory, leading to astructural theory for many classes of non-separable Banach spaces. Just to mention someadvances, we refer, e.g. , to [8], [18], [33], [45], [66], [72], [80]. Such a theory is tightlyconnected to differentiability [12], [27], [28], [30], [31], [42], classes of compacta [10], [14],[15], [16], [17], [32], [49], [56], combinatorics [6], [22], [23], [59], [63], [69], [77], [75].An important tool in the area was introduced by Fabian [24], who used Jayne–Rogersselectors [40] to show that every weakly countably determined Asplund Banach space isindeed WCG. Jayne–Rogers selectors were also deeply involved, together with Simons’lemma [70], [35], in the proof that the dual of every Asplund space admits a PRI, [26].The techniques of [24] also used ingredients from [42], where it is shown, among others,that WCG Banach spaces with a Fréchet smooth norm admit a shrinking M-basis. Resultsof this nature lead to the conjecture that Asplund Banach spaces with a norming M-basisare WCG. This question is originally due to Godefroy, who, at the times when [21] was
Date : July 29, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Norming Markuševič basis, Asplund Banach space, weakly compactly generatedBanach space, ̺ -function, semi-Eberlein compact space.P. Hájek was supported in part by OPVVV CAAS CZ.02.1.01/0.0/0.0/16_019/0000778.Research of T. Russo was supported by the project International Mobility of Researchers in CTUCZ.02.2.69/0.0/0.0/16_027/0008465 and by Gruppo Nazionale per l’Analisi Matematica, la Probabilità ele loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica (INdAM), Italy.J. Somaglia was supported by Università degli Studi di Milano, Research Support Plan 2019 and by GruppoNazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionaledi Alta Matematica (INdAM), Italy.Research of S. Todorčević is partially supported by grants from NSERC(455916) and CNRS(UMR7586). in preparation, conjectured that a similar use of Jayne–Rogers selectors might produce alinearly dense weakly compact subset, in presence of a norming M-basis. Problem 1.1 (G. Godefroy) . Let X be an Asplund space with a norming Markuševičbasis. Must X be weakly compactly generated?The problem was subsequently recorded in various articles and books, see, e.g. , [2], [38,p. 211], [36, Problem 112]. The main result of our paper is a negative answer to thisproblem, in the form of the following result. Theorem A.
There exists an Asplund space X with a -norming M-basis such that X isnot WCG.The proof of Theorem A will be given in Section 4.1. As a matter of fact, the M-basisthat we construct there is additionally an Auerbach basis, see Remark 4.3. Moreover, asmall elaboration over the argument also produces a counterexample with a long monotoneSchauder basis (Theorem 4.7).Since the result [5] that WCG Banach spaces admit an M-basis, and, therefore, reflexivespaces have a shrinking basis, it readily became clear that M-bases with additional prop-erties would have been instrumental in the characterisation of several classes of Banachspaces, [38, Chapter 6], [80]. In particular, it was natural to ask which class of Banachspaces is characterised by the presence of a norming M-basis. This lead to the famousquestion, due to John and Zizler, whether every WCG Banach space admits a normingM-basis [43], that was recently solved in the negative by the first-named author, [37]. Inthis sense, Problem 1.1 can be considered as a converse to the said John’s and Zizler’squestion.As it turns out, there is an elegant characterisation of Banach spaces that admit a shrink-ing M-basis, in the form of the following result, due to the efforts of many mathematicians,[24], [78], [62], [41], [42]. We refer to [38, Theorem 6.3], or [25, Theorem 8.3.3] for a proof. Theorem 1.2.
For a Banach space X , the following are equivalent: (i) X admits a shrinking M-basis; (ii) X is WCG and Asplund; (iii) X is WLD and Asplund; (iv) X is WLD and X ∗ has a dual LUR norm; (v) X is WLD and it admits a Fréchet smooth norm. Problem 1.1 is clearly closely related to this result, since it amounts to asking whetherthe assumption in (ii) that X is WCG could be replaced by the existence of a normingM-basis.Our construction in Theorem A heavily depends on the existence of a peculiar scatteredcompact space, whose properties we shall record in Theorem B below. Before its statement,we need one piece of notation.Given a set S , we identify the power set P ( S ) with the product { , } S , via the canonicalcorrespondence A ↔ A ( A ⊆ S ). Since { , } S is a compact topological space in its natural N ASPLUND SPACE WITH NORMING M-BASIS THAT IS NOT WCG 3 product topology, this identification allows us to introduce a compact topology on P ( S ) .Throughout our article, any topological consideration relative to P ( S ) will refer to the saidtopology, that we shall refer to as the product , or pointwise , topology. Theorem B.
There exists a family F ̺ ⊆ [ ω ] <ω of finite subsets of ω such that K ̺ := F ̺ has the following properties:(i) { α } ∈ K ̺ for every α < ω ,(ii) [0 , α ) ∈ K ̺ for every α ω ,(iii) if A ∈ K ̺ is an infinite set, then A = [0 , α ) for some α ω ,(iv) K ̺ is scattered.The subscript ̺ in our notation for the family F ̺ reflects the rôle of the choice of a ̺ -function, a rather canonical semi-distance on ω , in the construction of the family F ̺ . ̺ -functions were introduced in [74] for a study of the way Ramsey’s theorem fails in theuncountable context. They appeared already in Banach space constructions, see, e.g. , [7],[58], [59]. We refer to [76], [11], [13] for a detailed presentation of this theory and furtherapplications in several areas. ̺ -functions are also tightly related to construction schemes ,[77], that also proved very useful in non-separable Banach space theory, see, e.g. , [57].In conclusion to this section, we briefly describe the organisation of the paper. Section 2contains a revision of the notions from non-separable Banach space theory that are relevantto our paper. The proof of Theorem B, together with a quick revision of the necessaryresults concerning ̺ -functions, will be given in Section 3. Section 4 is independent fromthe argument in Section 3, as it only depends on the statement of Theorem B. Apart forthe proof of Theorem A, we observe there that the compact space in Theorem B also offersan interesting example for the theory of semi-Eberlein compacta, solving a problem from[52]. Finally, in Section 5, we discuss the main problem in the C ( K ) case, where K is anadequate compact; in particular, we show that C ( K ) has a -norming M-basis, whenever K is adequate. Finally, a typographical note: the symbol (cid:4) denotes the end of a proof,while, in nested proofs, we use (cid:3) for the end of the inner proof.2. Preliminary definitions and results
General conventions.
Our notation concerning Banach spaces is standard, as inmost textbooks in Banach space theory; we refer, e.g. , to [1], [29]. All our results in thepaper are valid for Banach spaces over either the real or the complex fields, with the sameproofs. By a subspace of a Banach space we understand a closed, linear subspace.We indicate by | S | the cardinality of a set S . For a set S and a cardinal number κ , wewrite [ S ] κ = { A ⊆ S : | A | = κ } and [ S ] <κ = { A ⊆ S : | A | < κ } . We denote by ω thefirst infinite ordinal and by ω the first uncountable one. We also adopt the convention toregard cardinal numbers as initial ordinals; hence, we write ω for ℵ , ω for ℵ , and so on.If A and B are subsets of an ordinal α , we write A < B meaning that a < b whenever a ∈ A and b ∈ B . Given a pair of ordinals α β , we denote by [ α, β ] and [ α, β ) the setscomprising all ordinals γ such that α γ β and α γ < β , respectively. We equipthe intervals [ α, β ] and [ α, β ) with the canonical order topology which turns every interval P. HÁJEK, T. RUSSO, J. SOMAGLIA, AND S. TODORČEVIĆ [ α, β ] into a compact space and makes [ α, β ) compact if and only if β is a successor ordinal.We follow this notation also for the intervals [0 , α ] and [0 , α ) . According to the standarddefinition of ordinals, the interval [0 , α ) coincides with the ordinal α ; however, we shallmostly keep the notation [0 , α ) to stress the topological structure rather than, say, the rôleof index set.Next, we shall record some basic notions concerning biorthogonal systems in Banachspaces. A biorthogonal sytem in a Banach space X is a system { x γ ; ϕ γ } γ ∈ Γ , with x γ ∈ X and ϕ γ ∈ X ∗ , such that h ϕ α , x β i = δ α,β ( α, β ∈ Γ ). A biorthogonal system is fundamental (or complete ) if span { x γ } γ ∈ Γ is dense in X ; it is total when span { ϕ γ } γ ∈ Γ is w ∗ -dense in X ∗ .A Markuševič basis (henceforth, M-basis) is a fundamental and total biorthogonal system.An
Auerbach basis is an M-basis such that k x γ k = k ϕ γ k = 1 , γ ∈ Γ .M-bases exist in many Banach spaces, in particular in every separable one, [60], andmany classes of Banach spaces admit characterisations in terms of M-bases (see, e.g. , [38,Chapter 6]). The simplest example of a Banach space without M-bases is ℓ ∞ , [44]; on theother hand, ℓ ∞ admits a fundamental biorthogonal system [19] (and, of course, also a totalone).An M-basis { x γ ; ϕ γ } γ ∈ Γ is shrinking if span { ϕ γ } γ ∈ Γ is dense in X ∗ . As we saw in Theorem1.2, this is a rather strong notion, as it implies that X is both Asplund and WCG. TheM-basis { x γ ; ϕ γ } γ ∈ Γ is λ -norming ( < λ ) if λ k x k sup (cid:8) |h ϕ, x i| : ϕ ∈ span { ϕ γ } γ ∈ Γ , k ϕ k (cid:9) ( x ∈ X ) , namely, if span { ϕ γ } γ ∈ Γ is a λ -norming subspace for X . { x γ ; ϕ γ } γ ∈ Γ is norming if it is λ -norming, for some λ > .An important example of a Banach space that admits no norming M-basis is C ([0 , ω ]) ,as proved by Alexandrov and Plichko [2]. On the other hand, C ([0 , ω ]) admits a strongand countably -norming M-basis, [2] (see also [38, Theorem 5.25]). Recall that an M-basis is countably λ -norming if { ϕ ∈ X ∗ : { γ ∈ Γ : h ϕ, x γ i 6 = 0 } is countable } is a λ -normingsubspace. Kalenda [47] proved that C ([0 , ω )) (that is isomorphic to C ([0 , ω ]) ) admits nocountably -norming M-basis. Moreover, C ([0 , ω ]) admits no countably norming M-basis,[50].2.2. Some classes of compact spaces.
A topological space K is scattered if every itsclosed subspace has an isolated point; in other words, K contains no non-empty perfectsubset. Every scattered compact is zero-dimensional , i.e. , it admits a basis consisting ofclopen sets [79, Theorem 29.7]. A compact space is countable if and only if it is metris-able and scattered [21, Lemma VI.8.2]; moreover, scattered compacta are closed undercontinuous images, [67].Given an arbitrary set Γ , the Σ -product of real lines is Σ(Γ) = { x ∈ [0 , Γ : |{ γ ∈ Γ : x ( γ ) = 0 }| ω } . (Here, and throughout the paper, we consider the product topology on the space [0 , Γ .)For an element x ∈ Σ(Γ) , we call the set supp( x ) := { γ ∈ Γ : x ( γ ) = 0 } the support of x . A topological space K is Fréchet–Urysohn if for every subset A of K and every p ∈ A , N ASPLUND SPACE WITH NORMING M-BASIS THAT IS NOT WCG 5 there exists a sequence in A that converges to p . It is easy to see that every Σ -product isFréchet–Urysohn, [49, Lemma 1.6]. It is also clear that Σ(Γ) is dense in [0 , Γ and it is countably closed (in the sense that the closure, in [0 , Γ , of every countable subset of Σ(Γ) is contained in
Σ(Γ) ).A compact space is
Eberlein if it is homeomorphic to a weakly compact subset of someBanach space. In their seminal paper [5], Amir and Lindenstrauss proved that everyEberlein compact is homeomorphic to a weakly compact subset of c (Γ) , for some set Γ .Let us also recall, in passing, that every Eberlein compact is homeomorphic to a weaklycompact subset of a reflexive Banach space, [18].A compact space K is Corson if K is homeomorphic to a subset of Σ(Γ) , for some set Γ .Therefore, Corson compacta are Fréchet–Urysohn; in particular, [0 , ω ] is not Corson. Ofcourse, every Eberlein compact is Corson. The converse implication is false, [14, Example5.1], [4], [72]; however, scattered Corson compacta are Eberlein, [3].A compact space is Valdivia if it is homeomorphic to a compact subspace K of [0 , Γ such that K ∩
Σ(Γ) is dense in K . If K is Valdivia and h : K → [0 , Γ is an embedding thatwitnesses this, namely if h ( K ) ∩ Σ(Γ) is dense in h ( K ) , we set Σ( K ) := h − (Σ(Γ)) . Σ( K ) is then called a Σ -subset of K and, by definition, it is dense in K . For further informationon Valdivia compacta we refer to the very detailed survey [49] and the references therein.We denote P (Γ) the power set of a set Γ , which, throughout our article, we identify withthe product { , } Γ , in the canonical way. This permits us to introduce a compact topologyon P (Γ) , that we also call the product, or pointwise, topology. With the said topology,it is easy to see that the correspondence α [0 , α ) defines an embedding of [0 , ω ] into P ( ω ) ; in particular, [0 , ω ] is a Valdivia compactum.As it turns out, [0 , ω ] is the archetypal example of a Valdivia compact space that is notCorson. Indeed, Deville and Godefroy [20] proved the very elegant result that a Valdiviacompact space is Corson if and only if it contains no copy of [0 , ω ] . Kalenda [48] (seealso [49, Proposition 3.12]) generalised this result as follows: if K ⊆ [0 , ω is such that K ∩ Σ( ω ) is dense in K and p ∈ K \ Σ( ω ) , there exists an embedding ϕ : [0 , ω ] → K with(i) ϕ ( α ) ∈ K ∩ Σ( ω ) , for α < ω ,(ii) supp( ϕ ( α )) ⊆ supp( ϕ ( β )) , for α < β ω ,(iii) ϕ ( ω ) = p .One more crucial property of Valdivia compacta is that they admit a ‘good’ system ofretractions, e.g. , [53]. The canonical way to build them is to use the following lemma,based on a rather standard closing-off argument; see, e.g. , [10, Lemma 1.2], [21, LemmaVI.7.5], or [46, Lemma 19.10]. For an element x ∈ [0 , Γ and J ⊆ Γ , we define x ↾ J by x ↾ J ( γ ) = ( x ( γ ) γ ∈ J γ / ∈ J. Lemma 2.1.
Let
K ⊆ [0 , Γ be a compact space such that K ∩
Σ(Γ) is dense in K . Forany infinite set I ⊆ Γ there exists a set J ⊆ Γ with I ⊆ J , | I | = | J | and such that x ∈ K = ⇒ x ↾ J ∈ K . P. HÁJEK, T. RUSSO, J. SOMAGLIA, AND S. TODORČEVIĆ
Non-separable Banach spaces.
A Banach space X is Asplund if every convexcontinuous function defined on a convex open subset U of X is Fréchet differentiable on adense G δ subset of U . It is well known that X is Asplund if and only if every separablesubspace of X has a separable dual, see [21, Theorem I.5.7]. In particular, we can see thatthe class of Asplund spaces is closed under taking subspaces and quotients. In the subclassof C ( K ) spaces, a C ( K ) space is Asplund if and only if the compact K is scattered, [61] (see, e.g. , [21, Lemma VI.8.3]). For further information and historical background on Asplundspaces we refer to [21, § 1.5], [25], [64], and the references therein.A Banach space is weakly compactly generated (WCG, for short) if it admits a linearlydense, weakly compact subset [5]; typical examples of WCG Banach spaces are separableBanach spaces, reflexive ones, c (Γ) for every index set Γ , or L ( µ ) , for a finite measure µ .If X is WCG, then the dual ball ( B X ∗ , w ∗ ) is an Eberlein compactum, [5]. The converseimplication however fails, in light of the famous Rosenthal’s example [66] of a non WCGsubspace of a WCG Banach space. There even are WCG Banach spaces with unconditionalbasis that contain non WCG subspaces (with unconditional bases), [9]. The situation ismore symmetric in the C ( K ) case, since C ( K ) is WCG if and only if K is Eberlein, if andonly if ( B C ( K ) ∗ , w ∗ ) is Eberlein, [5].A broader class of Banach spaces is constituted by WLD Banach spaces. A Banach space X is weakly Lindelöf determined (hereinafter, WLD) if the dual ball ( B X ∗ , w ∗ ) is Corson,[8]. From the stability properties of Corson compacta under subspaces and continuousimages, it readily follows that WLD spaces are closed under subspaces and quotients.As we already saw in Theorem 1.2, when restricted to Asplund spaces, WCG and WLDcollapse to the same notion. Therefore, for our paper, the subtlety of the distinctionbetween WCG, subspace of WCG and WLD will not be extremely relevant. A C ( K ) spaceis WLD if and only if K is Corson and it has property (M), namely every measure on K has separable support, [10].Incidentally, there results yield a, unnecessarily sophisticated, Banach space theoreticalproof of Alster’s result [3] that scattered Corson compacta are Eberlein. Indeed, everyscattered compact space has (M), since measures are even countably supported, [67]; hence, C ( K ) is WLD and Asplund when K is Corson and scattered. But then C ( K ) is WCG,whence K is Eberlein. 3. The proof of Theorem B
The goal of the present section is the proof of Theorem B. As we already mentionedin the Introduction, the construction of the family F ̺ depends upon the choice of a ̺ -function with some additional properties. Therefore, we start the section recalling somefacts concerning ̺ -functions.We will consider functions ̺ : [ ω ] → ω and it will be convenient to identify their domain [ ω ] with the set { ( α, β ) ∈ ω : α < β } . This just amounts to replacing the unordered pair { α, β } with the ordered one ( α, β ) , where α < β . This permits writing ̺ ( α, β ) instead of the more cumbersome ̺ ( { α, β } ) . It will N ASPLUND SPACE WITH NORMING M-BASIS THAT IS NOT WCG 7 also be useful to extend the domain of such functions by adding the boundary conditionthat ̺ ( α, α ) = 0 ( α < ω ). Throughout the section, we shall adhere to these conventions(which follow precisely those of [76]). Definition 3.1. A ̺ -function , or ordinal metric , on ω is a function ̺ : [ ω ] → ω with thefollowing properties:( ̺ { ξ α : ̺ ( ξ, α ) n } is a finite set, for every α < ω and n < ω ,( ̺ ) ̺ ( α, γ ) max { ̺ ( α, β ) , ̺ ( β, γ ) } for every α < β < γ < ω ,( ̺ ̺ ( α, β ) max { ̺ ( α, γ ) , ̺ ( β, γ ) } for every α < β < γ < ω .Several functions with the above properties, frequently originating as characteristics ofsome walk on ordinals, are constructed and studied in [76, Chapter 3]. Here we shall needthe existence of a ̺ -function on ω with the following properties. Proposition 3.2 ([76, Lemma 3.2.2]) . There exists a ̺ -function ̺ : [ ω ] → ω such that (i) ̺ ( α, β ) > for all α < β < ω , (ii) ̺ ( α, γ ) = ρ ( β, γ ) , for all α < β < γ < ω . Let us just mention that condition (i) is obvious from the definition of ̺ (given in [76,Definition 3.2.1]), while (ii) is the content of [76, Lemma 3.2.2].We are now in position to define the desired compact space K ̺ . Definition 3.3.
Let ̺ : [ ω ] → ω be a ̺ -function satisfying (i)-(ii) of Proposition 3.2.Given n < ω and α < ω , let F n ( α ) := { ξ α : ̺ ( ξ, α ) n } . Moreover, we denote F ̺ := { F n ( α ) : n < ω, α < ω } and K ̺ := F ̺ , where the closure is intended in the pointwise topology of P ( ω ) .Let us list in the next fact some properties of the sets F n ( α ) that are immediate conse-quence of their definition. Fact 3.4.
For every α < ω the following hold: (i) F n ( α ) ⊆ [0 , α ] for all n < ω , (ii) F ( α ) = { α } , (iii) F k ( α ) ⊆ F n ( α ) for k n < ω , (iv) | F n ( α ) | n + 1 for every n < ω , (v) the sequence ( F n ( α )) n<ω converges to [0 , α ] . Concerning the proof of these assertions, we just note that, according to Proposition3.2(ii), ̺ ( · , α ) defines an injection of [0 , α ] into ω ; whence (iv) follows. Let us also observethat (iv) is a uniform version of condition ( ̺ K ̺ defined above satisfies the conditions stated in Theorem B. For convenience ofthe reader, we shall repeat here the statement of the result under consideration. P. HÁJEK, T. RUSSO, J. SOMAGLIA, AND S. TODORČEVIĆ
Theorem 3.5.
The compact space K ̺ defined above has the following properties: (i) { α } ∈ K ̺ for every α < ω , (ii) [0 , α ) ∈ K ̺ for every α ω , (iii) if A ∈ K ̺ is an infinite set, then A = [0 , α ) for some α ω , (iv) K ̺ is scattered. Due to our identification between P ( ω ) and { , } ω , we can see F ̺ as a subset of Σ( ω ) .Since F ̺ is dense in K ̺ , we derive that K ̺ is Valdivia. We will see later in Section 4.3 thatit is even semi-Eberlein.As the reader will see, the main part of the proof, where we use the properties of thefunction ̺ , consists in establishing (iii). Then (iv) is consequence of (iii), while (i) and (ii)are immediate. Proof. (i) is obvious, since { α } = F ( α ) ∈ F ̺ for every α < ω , in light of Fact 3.4(ii).(ii) Pick α ω arbitrarily. If α = α ′ + 1 is a successor ordinal, then [0 , α ) = [0 , α ′ ] =lim n<ω F n ( α ′ ) , by Fact 3.4(v). Thus, [0 , α ) ∈ K ̺ . If α > is limit, then [0 , α ) is thelimit of the net ([0 , β + 1)) β<α , whence [0 , α ) ∈ K ̺ in this case as well. Finally, if α = 0 , [0 , α ) = ∅ = lim j<ω { j } ∈ K ̺ .(iii) Let A ∈ K ̺ be an infinite set. Our goal is to show that α ∈ A, ˜ α < α = ⇒ ˜ α ∈ A. We shall start with the case when A is a countable set. Claim 3.6.
Let A ∈ K ̺ be such that | A | = ω . If α ∈ A and ˜ α < α , then ˜ α ∈ A .Proof of Claim 3.6. Pick A ∈ K ̺ such that | A | = ω and fix α ∈ A and ˜ α < α . Since A is countable, it belongs to the Σ -subspace Σ( ω ) , which is Fréchet–Urysohn. Therefore,there exists a sequence ( F n k ( α k )) k<ω ⊆ F ̺ such that F n k ( α k ) → A . Observe that, A being infinite, it cannot belong to F ̺ ; thus, we can select the sequence ( F n k ( α k )) k<ω to beinjective. Moreover, up to discarding finitely many terms from the sequence, we can alsoassume that α ∈ F n k ( α k ) for each k < ω (since α ∈ A and F n k ( α k ) → A ).We next observe that the sequence ( n k ) k<ω is necessarily unbounded. Indeed, if thiswere not the case, by Fact 3.4(iv) there would exist M < ω such that | F n k ( α k ) | M foreach k < ω . But then, it would follow that | A | M as well, contrary to our assumption.Consequently, up to passing to a subsequence and relabelling, we can also assume that ̺ ( ˜ α, α ) n k for each k < ω .The condition α ∈ F n k ( α k ) yields that α α k and ̺ ( α, α k ) n k . Therefore, ˜ α < α α k and property ( ̺ ) imply that ̺ ( ˜ α, α k ) max { ̺ ( ˜ α, α ) , ̺ ( α, α k ) } n k . Consequently, we obtain ˜ α ∈ F n k ( α k ) for every k < ω , which implies ˜ α ∈ A , as wedesired. (cid:3) Finally, we shall consider the case when A is uncountable and we shall show that A =[0 , ω ) . Towards a contradiction, assume that there exists α < ω such that α / ∈ A . Since N ASPLUND SPACE WITH NORMING M-BASIS THAT IS NOT WCG 9 A is uncountable, we can also pick β ∈ A , β > α , such that A ∩ [0 , β ] is an infinite set(which might not belong to K ̺ , though). However, in light of Lemma 2.1, there exists γ < ω , γ > β , such that A ∩ [0 , γ ) ∈ K ̺ . Therefore, A ∩ [0 , γ ) ∈ K ̺ is a set of cardinality ω that is not an initial interval, since α / ∈ A ∩ [0 , γ ) , while β ∈ A ∩ [0 , γ ) . This contradictsClaim 3.6 above and concludes the proof of (iii).(iv) Let D be any closed subset of K ̺ . We shall show that D admits an isolated point. Inthe case that every element of D is an initial interval (namely, of the form [0 , α ) , for some α < ω ), then D is homeomorphic to a closed subset of [0 , ω ] and, therefore, it containsan isolated point. Consequently, we can assume that there is D ∈ D that is not an initialinterval. Thus, we can pick α < β < ω such that α / ∈ D and β ∈ D .Consider the partially ordered set P := { D ∈ D : α / ∈ D, D ⊆ D } , partially ordered by inclusion. Every chain in such a partially ordered set admits an upperbound given by the union of its elements; indeed, such union belongs to D , D being aclosed subset of K ̺ . By Zorn’s lemma, we can pick a maximal element M ∈ P . Since α / ∈ M and β ∈ M , M is not an initial interval; hence, according to (iii), it is a finite set.Consequently, the set U := { D ∈ D : α / ∈ D, M ⊆ D } , is an open neighbourhood of M in D . However, by maximality of M , U = { M } , whichshows that M ∈ D is the desired isolated point. (cid:4) Remark . The assertion, in (iii), that the unique uncountable set in K ̺ is [0 , ω ) canalso be proved in a different way using, instead of Lemma 2.1, Kalenda’s extension [48] ofDeville–Godefroy’s theorem [20] that we mentioned already in Section 2. Indeed, as | A | = ω , there exists a continuous injection ϕ : [0 , ω ] → K ̺ such that ϕ ( ω ) = A , ϕ ( α ) ⊆ ϕ ( β ) whenever α < β ω and | ϕ ( α ) | max {| α | , ω } . If we set A α := ϕ ( α ) ( α < ω ) , ( A α ) α<ω is a non-decreasing family of sets such that ∪ α<ω A α = A (due to the continuity of ϕ )and | A α | ω . Therefore, there exists α < ω such that A α is an infinite set, whenever α α < ω . By Claim 3.6, every such A α is an initial interval, whence A is an initialinterval as well.4. Theorem A and other consequences of Theorem B
Proof of Theorem A.
This section is dedicated to the proof of our main result.
Proof of Theorem A.
According to Theorem B, we can pick a family F ̺ ⊆ [ ω ] <ω such thatthe compact K ̺ := F ̺ ⊆ P ( ω ) has the following properties:(i) { α } ∈ K ̺ for every α < ω ,(ii) [0 , α ) ∈ K ̺ for every α ω ,(iii) if A ∈ K ̺ is an infinite set, then A = [0 , α ) for some α ω ,(iv) K ̺ is scattered. We define a biorthogonal system { f γ ; µ γ } γ<ω in the Banach space C ( K ̺ ) as follows. For γ < ω , let f γ ∈ C ( K ̺ ) f γ ( A ) = ( γ ∈ A γ / ∈ A ( A ∈ K ̺ ) µ γ := δ { γ } ∈ M ( K ̺ ) µ γ ( S ) = ( { γ } ∈ S { γ } / ∈ S ( S ⊆ K ̺ ) . Note that, by (i), { γ } ∈ K ̺ for each γ < ω , whence each µ γ is, indeed, a measure on K ̺ . Moreover, f γ is a continuous function on K ̺ , since the set { A ∈ K ̺ : γ ∈ A } isclearly clopen. Finally, h µ γ , f α i = h δ { γ } , f α i = f α ( { γ } ) = δ α,γ . Therefore, { f γ ; µ γ } γ<ω is awell-defined biorthogonal system in C ( K ̺ ) .We are now in position to define the Banach space X ̺ := span { f γ } γ<ω ⊆ C ( K ̺ ) . Weshall show that X ̺ is the Banach space we are seeking, namely that X ̺ is an Asplund spacewith a -norming M-basis and that X ̺ is not WLD. Some of these properties are actuallyobvious. Indeed, X ̺ is an Asplund space, being a subspace of the Asplund space C ( K ̺ ) ,by (iv). Moreover, the biorthogonal system { f γ ; µ γ } γ<ω naturally induces a biorthogonalsystem { f γ ; µ γ ↾ X ̺ } γ<ω on X ̺ . Such a system is clearly a fundamental (in the sense that { f γ } γ<ω is linearly dense in X ̺ ) biorthogonal system. Claim 4.1. X ̺ is not WLD.Proof of Claim 4.1. We shall show that the dual ball ( B X ̺ ∗ , w ∗ ) is not Corson, by provingthat [0 , ω ] embeds therein. From condition (iii) we infer that β [0 , β ) defines anembedding ι of [0 , ω ] into K ̺ ; therefore, it suffices to prove that ( B X ̺ ∗ , w ∗ ) contains ahomeomorphic copy of K ̺ . This is actually a standard consequence of the fact that thefunctions { f γ } γ<ω separate points of K ̺ . Let us give the details below.It is well known that every compact space K embeds in ( B M ( K ) , w ∗ ) via the map δ : K → ( B M ( K ) , w ∗ ) given by p δ p ( p ∈ K ). Moreover, we can consider the w ∗ - w ∗ -continuousquotient map q : M ( K ̺ ) → X ̺ ∗ defined by µ µ ↾ X ̺ . Hence, the function e := q ◦ δ : K ̺ → ( B X ̺ ∗ , w ∗ ) , namely the function given by the rule A δ A ↾ X ̺ ( A ∈ K ̺ ), is continuous. [0 , ω ] (cid:31) (cid:127) ι / / K ̺ (cid:31) (cid:127) δ / / (cid:22) v e ( ( ❘❘❘❘❘❘❘❘ ( B M ( K ̺ ) , w ∗ ) q (cid:15) (cid:15) ( B X ̺ ∗ , w ∗ ) We shall show that e is injective, which, due to the compactness of K ̺ , implies that it isthe desired homeomorphic embedding. Given distinct A, B ∈ K ̺ , pick γ ∈ A ∆ B ; assume,for example, γ ∈ A \ B . Then h δ A ↾ X ̺ , f γ i = h δ A , f γ i = f γ ( A ) = 1 , while h δ B ↾ X ̺ , f γ i = f γ ( B ) = 0 . Hence, δ A ↾ X ̺ = δ B ↾ X ̺ , as desired. (cid:3) N ASPLUND SPACE WITH NORMING M-BASIS THAT IS NOT WCG 11
In order to conclude the proof we thus just need to prove that the biorthogonal sys-tem { f γ ; µ γ ↾ X ̺ } γ<ω is -norming for X ̺ . Note that this implies, in particular, that span { µ γ ↾ X ̺ } γ<ω is w ∗ -dense in X ̺ ∗ . We start with the following observation. Claim 4.2.
Let A ∈ F ̺ . Then (4.1) δ A ↾ X ̺ = X α ∈ A δ { α } ↾ X ̺ . In particular, it follows that(4.2) { δ A ↾ X ̺ : A ∈ F ̺ } ⊆ span { µ γ ↾ X ̺ } γ<ω . Proof of Claim 4.2.
Recall that every element A ∈ F ̺ is a finite set and { α } ∈ F ̺ , forevery α ∈ A ; thus, the right hand side in (4.1) is a well-defined functional on X ̺ . Ofcourse, it is sufficient to show that the functional δ A ↾ X ̺ − P α ∈ A δ { α } ↾ X ̺ vanishes on thelinearly dense set { f γ } γ<ω . Fix γ < ω ; by definition of f γ , we have *X α ∈ A δ { α } , f γ + = X α ∈ A f γ ( { α } ) = X α ∈ A δ γ,α = ( γ ∈ A γ / ∈ A = f γ ( A ) = h δ A , f γ i . (cid:3) Finally, for every f ∈ X ̺ we have k f k = max A ∈K ̺ | f ( A ) | = sup A ∈F ̺ | f ( A ) | = sup A ∈F ̺ |h δ A , f i| ( . ) sup n |h µ, f i| : µ ∈ span { µ γ ↾ X ̺ } γ<ω , k µ k o . This shows that the M-basis { f γ ; µ γ ↾ X ̺ } γ<ω is -norming for X ̺ and concludes the proof. (cid:4) Remark . As it is clear from the proof, the M-basis { f γ ; µ γ ↾ X ̺ } γ<ω satisfies k f γ k = k µ γ ↾ X ̺ k = 1 for each γ < ω ; in other words, it is additionally an Auerbach basis.4.2. Further properties of the space X ̺ . In this section we shall prove the resultmentioned in the Introduction that the Banach space X ̺ can also be assumed to have a longmonotone Schauder basis. The argument will be a simple modification of the constructionin Section 4.1; in particular, we shall continue to denote { f γ ; µ γ } γ<ω the biorthogonalsystem introduced there.The main idea is that the Banach space X Λ := span { f γ } γ ∈ Λ shares the main featuresof X ̺ , whenever Λ is an uncountable subset of ω . On the other hand, it is a folkloreresult that if { f γ ; µ γ } γ<ω is a -norming M-basis for a Banach space X , then there existsan uncountable subset Λ of ω such that { f γ } γ ∈ Λ is a monotone long Schauder basis (seeFact 4.4 below).For the definitions of long Schauder bases, long (Schauder) basic sequences and their basisconstants we shall refer, e.g. , to [38, § 4.1], or [71, § 17]. Here, we shall restrict ourselves tospelling out the following useful characterisation, [71, Theorem 17.8]. A collection ( e γ ) γ< Γ of non-zero vectors in a Banach space X is a long basic sequence if and only if there existsa constant C > such that k y k C k y + z k whenever y ∈ span { e γ } γ< Ω , z ∈ span { e γ } Ω γ< Γ , and Ω < Γ (and, in this case, the basisconstant of ( e γ ) γ< Γ is at most C ).We are now in position to state the following folklore fact, based on Mazur’s technique(compare with [38, Corollary 4.11]). Fact 4.4.
Let { f γ ; µ γ } γ<ω be a θ -norming M-basis for a Banach space X . Then thereexists an uncountable subset Λ of ω such that { f γ } γ ∈ Λ is a long basic sequence in X (inthe natural ordering induced on Λ by Γ ) with basis constant at most /θ .Proof. If Ω is any countable subset of ω , there is a countable subset S Ω of ω such that θ k x k sup {|h µ, x i| : µ ∈ span { µ γ } γ ∈ S Ω , k µ k } , for every x ∈ span { f γ } γ ∈ Ω . Indeed, let ( g j ) j<ω be a dense sequence in span { f γ } γ ∈ Ω and,for each j < ω , find a sequence ( µ jk ) k<ω ∈ span { µ γ } γ<ω , k µ jk k , such that θ k g j k sup k<ω |h µ jk , g j i| . Then, any countable set S Ω ⊆ ω such that ( µ jk ) k,j<ω ⊆ span { µ γ } γ ∈ S Ω is as desired.Therefore, a standard transfinite induction argument gives an uncountable subset Λ =( λ ξ ) ξ<ω of ω , where the ordinals λ ξ are enumerated in increasing order, and an increasingfamily ( S ξ ) ξ<ω of countable subsets of ω such that S ξ < λ ξ and θ k x k sup (cid:8) |h µ, x i| : µ ∈ span { µ γ } γ ∈ S ξ , k µ k (cid:9) , whenever x ∈ span { f λ γ } γ<ξ .Consequently, for every ξ < ω and every x ∈ span { f λ γ } γ<ξ and y ∈ span { f λ γ } ξ γ<ω wehave y ∈ ker µ γ ( γ ∈ S ξ ); hence, θ k x k sup (cid:8) |h µ, x i| : µ ∈ span { µ γ } γ ∈ S ξ , k µ k (cid:9) = sup (cid:8) |h µ, x + y i| : µ ∈ span { µ γ } γ ∈ S ξ , k µ k (cid:9) k x + y k . By the characterisation mentioned above, we derive that ( f γ ) γ ∈ Λ is the desired long basicsequence with basis constant at most /θ . (cid:4) Remark . We stated the result for M-bases of length ω since we shall only need thisparticular case; however, a standard modification of the argument also proves the followingmore general facts. If Γ is an uncountable ordinal and { f γ ; µ γ } γ< Γ is a θ -norming M-basisfor X , there exist a set Λ ⊆ Γ with | Λ | = | Γ | and a well ordering of Λ such that ( f γ ) γ ∈ Λ isa long basic sequence in X in the ordering. In case Γ is a regular cardinal, the orderingcan be chosen to be the one induced by Γ . Finally, if Γ is countable, for every ε > , thereexists an increasing sequence ( γ j ) j<ω in Γ such that ( f γ j ) j<ω is a basic sequence with basisconstant at most /θ + ε . N ASPLUND SPACE WITH NORMING M-BASIS THAT IS NOT WCG 13
Next, we shall observe the obvious fact that passing to a subset of the index set of anorming M-basis produces a norming M-basis for the corresponding subspace.
Fact 4.6.
Let { f γ ; µ γ } γ< Γ be a θ -norming M-basis for a Banach space X and let Λ ⊆ Γ .Set X Λ := span { f γ } γ ∈ Λ . Then { f γ ; µ γ ↾ X Λ } γ ∈ Λ is a θ -norming M-basis for X Λ .Proof. Obviously, { f γ ; µ γ ↾ X Λ } γ ∈ Λ is a fundamental biorthogonal system for X Λ . If µ ∈ span { µ γ } γ< Γ , write µ = P γ< Γ a γ µ γ , where only finitely many scalars a γ are non-zero.Then µ ↾ X Λ = X γ ∈ Λ a γ µ γ ↾ X Λ . Therefore, when x ∈ X Λ , we have θ k x k sup (cid:8) |h µ ↾ X Λ , x i| : µ ∈ span { µ γ } γ< Γ , k µ k (cid:9) = sup (cid:8) |h µ, x i| : µ ∈ span { µ γ ↾ X Λ } γ ∈ Λ , k µ k (cid:9) . Thus, span { µ γ ↾ X Λ } γ ∈ Λ is θ -norming for X Λ , as desired. (cid:4) We are now ready to state and prove the announced result.
Theorem 4.7.
There exists an Asplund space X with a -norming Auerbach basis { f γ ; µ γ } γ<ω such that ( f γ ) γ<ω is a long monotone Schauder basis and yet X is not WCG.Proof. Consider the Banach space X ̺ with -norming M-basis { f γ ; µ γ } γ<ω constructedin Section 4.1. Recall that k f γ k = k µ γ k = 1 ( γ < ω ), see Remark 4.3. In the lightof Fact 4.4, there exists an uncountable subset Λ of ω such that ( f γ ) γ ∈ Λ is a monotonelong Schauder basis for X Λ := span { f γ } γ ∈ Λ . Moreover, Fact 4.6 yields that X Λ admits a -norming Auerbach basis, given by { f γ ; µ γ ↾ X Λ } γ ∈ Λ . Since X Λ ⊆ X ̺ , it is also clear that X Λ is Asplund. Consequently, it only remains to prove that the space X Λ is not WCG.In order to achieve that, we shall show that [0 , ω ] embeds into ( B X Λ ∗ , w ∗ ) ; hence, ( B X Λ ∗ , w ∗ ) is not Corson and X Λ is not WLD. Let us enumerate Λ as an increasing trans-finite sequence ( λ ξ ) ξ<ω . We then consider the continuous function π : [0 , ω ] → ( B X Λ ∗ , w ∗ ) defined by π ( α ) := δ [0 ,α ) ↾ X Λ ; observe that for α < ω and λ ∈ Λ we have h π ( α ) , f λ i = h δ [0 ,α ) ↾ X Λ , f λ i = f λ ([0 , α )) . Then, using the density of span { f λ } λ ∈ Λ in X Λ , we obtain:(i) π ( α ) = π ( λ ξ +1 ) if α ∈ ( λ ξ , λ ξ +1 ] ;(ii) π ( α ) = π ( λ ) if α ∈ [0 , λ ] .(iii) π ( α ) = π ( λ ξ ) if ξ is a limit ordinal and α ∈ [sup β<ξ λ β , λ ξ ] .We observe that, if ξ < ξ < ω , then h π ( λ ξ ) , f λ ξ i = f λ ξ ([0 , λ ξ )) = 0 , while h π ( λ ξ ) , f λ ξ i = f λ ξ ([0 , λ ξ )) = 1 . Therefore, π ( λ ξ ) = π ( λ ξ ) . Consequently, the map h : [0 , ω ] → ( B X Λ ∗ , w ∗ ) defined by h ( ξ ) = ( π ( λ ξ ) , ξ < ω ,π ( ω ) , ξ = ω . is a injection into ( B X Λ ∗ , w ∗ ) . Let us show that h is also continuous. Indeed, let γ ∈ [0 , ω ] be a limit ordinal and let { γ η } be a net converging to γ = sup γ η . In case γ < ω , then by(iii) we have π (sup λ γ η ) = π ( λ γ ) , which, by the continuity of π , yields lim h ( γ η ) = h ( γ ) . On the other hand, when γ = ω , the continuity of the map π ensures that lim h ( γ η ) = h ( ω ) .Therefore [0 , ω ] embeds into ( B X Λ ∗ , w ∗ ) . (cid:4) Answering a question of Argyros, it was proved in [58] that there exists a c -saturated,non-separable Banach space X that contains no unconditional long basic sequence. In par-ticular, every infinite-dimensional subspace of X contains an unconditional basic sequence.Thus, the Banach space X exhibits a radical discrepancy between the behaviour of separa-ble and non-separable subspaces. The argument in [58] also heavily uses the machinery of ̺ -functions—differently from how it is done in our proof—combined with techniques orig-inating from Schlumprecht’s construction of an arbitrary distortable Banach space, [68].Let us also refer to [7], [11, Chapter A.6], and [76, § 3.5] for a related construction.We do not know if the Banach space X ̺ is also a solution to Argyros’ question, namely,we do not know if X ̺ contains unconditional long basic sequences. (Notice that X ̺ is c -saturated, being a subspace of C ( K ̺ ) , where K ̺ is scattered; see, e.g. , [29, Theorem 14.26].)In particular, we do not know if c ( ω ) embeds in X ̺ . However, we shall show the weakerfact that no uncountable subset of the vectors of the M-basis { f γ ; µ γ } γ<ω can be anunconditional long basic sequence in X ̺ . Proposition 4.8.
Let { f γ ; µ γ } γ<ω be the -norming M-basis for the Banach space X ̺ . If Λ ⊆ ω is uncountable, then ( f γ ) γ ∈ Λ is not an unconditional long basic sequence.Proof. Towards a contradiction, assume that there exists an uncountable subset Λ of ω such that ( f γ ) γ ∈ Λ is unconditional. Then, the Asplund space X Λ := span { f γ } γ ∈ Λ admitsa long unconditional basis, whence it follows that { f γ ; µ γ ↾ X Λ } γ ∈ Λ is a shrinking M-basis([39], e.g. , [38, Theorem 7.39]). Consequently, it follows that X Λ is WCG. However, this isnot the case, as we saw in the proof of Theorem 4.7. (cid:4) Weak P-points in semi-Eberlein compacta.
In this part we shall observe thatthe compact space constructed in Theorem B also provides an interesting example in thetheory of semi-Eberlein compact spaces. Semi-Eberlein compacta were introduced by Kubiśand Leiderman in [52], as a natural weakening of the definition of Eberlein compact. Moreprecisely, the definition of a semi-Eberlein compact space originates from the definition ofEberlein compact by the same generalisation that leads to Valdivia compacta from Corsonones. The formal definition reads as follows.
Definition 4.9.
A compact space is semi-Eberlein if it is homeomorphic to a compactspace
K ⊆ [0 , Γ such that K ∩ c (Γ) is dense in K .A compact space is ... if it is homeomorphic to K ⊆ [0 , Γ such that ... Valdivia semi-Eberlein ? _ o o Corson ?(cid:31) O O Eberlein ?(cid:31) O O ? _ o o K ∩
Σ(Γ) is dense in K K ∩ c (Γ) is dense in K ? _ o o K ⊆
Σ(Γ) ?(cid:31) O O K ⊆ c (Γ) ?(cid:31) O O ? _ o o N ASPLUND SPACE WITH NORMING M-BASIS THAT IS NOT WCG 15
Obviously, every Eberlein compact is semi-Eberlein and every semi-Eberlein is Valdivia.The Tikhonov cube [0 , ω (or, more generally, every non Corson adequate compact, seeSection 5 below) is a typical example of a semi-Eberlein compact that is not Corson (and,in particular, not Eberlein). In order to offer an example of a Valdivia compact that is notsemi-Eberlein, we need to recall the following notion. Definition 4.10 ([34], [54]) . A point p in a topological space X is a P-point if p isnot isolated and for every countable family ( U j ) j<ω of neighbourhoods of p , ∩ j<ω U j is aneighbourhood of p . A point p ∈ X is said a weak P-point if p is not isolated and it is limitpoint of no countable set in X \ { p } .An important result due to Kubiś and Leiderman ([52, Theorem 4.2]) is the fact thatsemi-Eberlein compacta do not admit P-points. Since perhaps the simplest example of a P-point is the point ω in the compact space [0 , ω ] , it follows that [0 , ω ] is not semi-Eberlein.The said result, combined with a forcing argument, also yields that the Corson compactconstructed in [73] is not semi-Eberlein, [52, Example 5.5]. These results motivated thequestion whether semi-Eberlein compacta can admit weak P-points, [52, Question 6.1]. Itis fairly easy to see that Theorem B also provides a positive answer to the above question. Proposition 4.11.
Let K ̺ be any compact space as in Theorem B. Then K ̺ is a semi-Eberlein compact space and it admits a weak P-point.Proof. K ̺ is semi-Eberlein, as witnessed by the dense subset F ̺ , that consists of finitesubsets of ω . Moreover, [0 , ω ) is a weak P-point in K ̺ . Indeed, since every set in K ̺ \{ [0 , ω ) } is countable, it follows that K ̺ \{ [0 , ω ) } = K ̺ ∩ Σ( K ̺ ) is countably closed. (cid:4) Adequate compacta and norming M-bases
In conclusion to our paper, we shall briefly discuss the main problem in the case of a C ( K ) space, where K is an adequate compact. We note the easy fact that every scatteredadequate compact is Eberlein, which, in particular, gives a positive answer to Godefroy’squestion in the realm of C ( K ) spaces, K adequate. We also show that C ( K ) has a -normingM-basis, whenever K is adequate.Recall that, given a set S , a family A ⊆ P ( S ) is adequate [72] if:(i) { x } ∈ A for each x ∈ S ;(ii) if A ∈ A and B ⊆ A , then B ∈ A ;(iii) if B ⊆ S is such that all finite subsets of B belong to A , then B ∈ A .Every adequate family A ⊆ P ( S ) is a closed subset of P ( S ) . When an adequate familyis considered as a compact space, it is customary to denote it K A and call it an adequatecompact .Let K A be a scattered adequate compact; in order to see that K A is Eberlein, we justneed to show that every A ∈ A is a finite set. If this is not the case and A ∈ A is infinite,then, by (ii), P ( A ) ⊆ A . However, P ( A ) is perfect, a contradiction. In other words, anadequate compact is scattered if and only if it is strong Eberlein ( cf . [38, Lemma 2.53]). Theorem 5.1.
Let K A be an adequate compact. Then C ( K A ) has a 1-norming M-basis. Proof.
Let F A = { A ∈ A : | A | < ω } , a dense subset of K A . For Γ ∈ F A , set f Γ ( A ) = ( if Γ ⊆ A, otherwise ,µ Γ = | Γ | X n =0 ( − | Γ |− n X ∆ ∈ [Γ] n δ ∆ . We shall show that the family { f Γ ; µ Γ } Γ ∈F A is the desired -norming M-basis (note that f Γ is continuous, since Γ is a finite set).We start by showing that { f Γ , µ Γ } Γ ∈F A is a biorthogonal system, that is h µ Γ , f Γ i = 1 if Γ = Γ and h µ Γ , f Γ i = 0 elsewhere. Indeed, assume Γ = Γ . Then, if n | Γ | and ∆ ∈ [Γ ] n , we have f Γ (∆) = 0 if and only if n = | Γ | and ∆ = Γ . Therefore, h µ Γ , f Γ i = f Γ (Γ ) = 1 . By the same argument, h µ Γ , f Γ i = 0 whenever Γ * Γ .On the other hand, suppose that Γ ⊆ Γ and Γ = Γ . Then, for every n < | Γ | andevery ∆ ∈ [Γ ] n we have f Γ (∆) = 0 . Moreover, when n > | Γ | , we have |{ ∆ ∈ [Γ ] n : f Γ (∆) = 0 }| = |{ ∆ ∈ [Γ ] n : Γ ⊆ ∆ }| = (cid:18) | Γ | − | Γ | n − | Γ | (cid:19) . Therefore, h µ Γ , f Γ i = | Γ | X n = | Γ | ( − | Γ |− n X ∆ ∈ [Γ ] n f Γ (∆)= | Γ | X n = | Γ | ( − | Γ |− n (cid:18) | Γ | − | Γ | n − | Γ | (cid:19) = | Γ |−| Γ | X n =0 ( − | Γ |−| Γ |− n (cid:18) | Γ | − | Γ | n (cid:19) = 0 , where the last equality depends on the binomial theorem (and the fact that | Γ | −| Γ | > ).Next, we show that span { f Γ } Γ ∈F A is dense in C ( K A ) . Since f ∅ ≡ ∈ span { f Γ } Γ ∈F A and { f Γ } Γ ∈F A separates points of K A , by the Stone–Weierstraß theorem, it is enough to provethat span { f Γ } Γ ∈F A is a subalgebra of C ( K A ) . For Γ , Γ ∈ F A , we have f Γ · f Γ ( A ) = ( if Γ ∪ Γ ⊆ A, otherwise . Therefore, if Γ ∪ Γ ∈ F A , then f Γ · f Γ = f Γ ∪ Γ ∈ span { f Γ } Γ ∈F A . On the other hand, if Γ ∪ Γ / ∈ F A , there is no A ∈ K A with Γ ∪ Γ ⊆ A . Hence, f Γ · f Γ ≡ . In either case, f Γ · f Γ ∈ span { f Γ } Γ ∈F A , whence span { f Γ } Γ ∈F A is a subalgebra of C ( K A ) .Finally, we show that span { µ Γ } Γ ∈F A is a -norming subspace, namely (by the Hahn–Banach theorem) that span { µ Γ } Γ ∈F A ∩ B M ( K A ) is w ∗ -dense in B M ( K A ) . Since F A is dense N ASPLUND SPACE WITH NORMING M-BASIS THAT IS NOT WCG 17 in K A , it suffices to show that { δ Γ } Γ ∈F A ⊆ span { µ Γ } Γ ∈F A . Indeed, we show by inductionthat { δ Γ } Γ ∈F A , | Γ | n ⊆ span { µ Γ } Γ ∈F A , for every n < ω . • For n = 0 , we just have δ ∅ = µ ∅ ∈ span { µ Γ } Γ ∈F A . • Inductively, suppose that n > and { δ Γ } Γ ∈F A , | Γ | n − ⊆ span { µ Γ } Γ ∈F A . If Γ ∈ F A is such that | Γ | = n , then δ Γ = µ Γ − n − X j =0 ( − n − j X ∆ ∈ [Γ] j δ ∆ ∈ span { µ Γ } Γ ∈F A , by the inductive assumption.Therefore, { f Γ ; µ Γ } Γ ∈F A is a 1-norming M-basis for C ( K A ) , as desired. (cid:4) Remark . We denote by σ (Γ) the one-point compactification of the discrete set Γ . Itis fairly easy to see that σ (Γ) ω is (homeomorphic to) an adequate compact, see, e.g. , [65].Therefore, our previous result generalises, with a similar (but cleaner) proof, [37, Theorem3], where a -norming M-basis is constructed in C ( σ (Γ) ω ) .Moreover, let us observe that in case K A is scattered, namely every set in A is finite, theM-basis constructed in the proof of Theorem 5.1 is even shrinking, as it is not hard to seefrom the above argument. This is, of course, in complete accordance with Theorem 1.2.In conclusion to our note, let us recall the classical result that C (2 ω ) has no unconditionalbasis, [51]. Here, ω denotes the Cantor set that, in our notation, is merely P ( ω ) , anadequate compact. In particular, the M-basis constructed in Theorem 5.1 is, in general,not unconditional. Acknowledgements.
The authors wish to express their gratitude to Marián Fabianand Gilles Godefroy for their insightful remarks on the problem considered in the article.Such remarks were extremely useful when preparing the final version of the manuscript.
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Department of Mathematics, Faculty of Electrical Engineering, Czech Tech-nical University in Prague, Technická 2, 166 27 Prague 6, Czech Republic
E-mail address : [email protected] (T. Russo) Department of Mathematics, Faculty of Electrical Engineering, Czech Tech-nical University in Prague, Technická 2, 166 27 Prague 6, Czech Republic
E-mail address : [email protected] (J. Somaglia) Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano,Via Cesare Saldini 50, 20133 Milano, Italy
E-mail address : [email protected] (S. Todorčević) Department of Mathematics, University of Toronto, Toronto, OntarioM5S 2E4, Canada and Institut de Mathématiques de Jussieu, Paris, France
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