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Descriptive Compact Spaces and Renorming Studia Math DESCRIPTIVE COMPACT SPACES AND RENORMING STUDIA MATH. 165(1) (2004), 39 – 52. L. ONCINA AND M. RAJA ABSTRACT. In this paper we study the class of descriptive compact spaces, the Banach spaces generated by descriptive compact subsets and their relation with renorming prob- lems. 1. INTRODUCTION Compact Hausdorff spaces which are fragmentable by a finer metric have been studied by many authors, see the book [7] for an account of this class of compacta. In this paper we shall consider a subclass of fragmentable compact spaces [25], namely D, that will allow us to construct an equivalent dual strictly convex norm on a dual Banach space X∗ if the dual unit ball lies in D. We have to introduce some terminology. Let {Hi : i ∈ I} be a family of subsets of a topological space (X,τ). The family is said to be isolated if it is discrete in its union endowed with the relative topology, that is, if for every i ∈ I and every x ∈ Hi, there is a τ-neighbourhood U of x such that Hj ∩ U = ∅ for every j ∈ I,j 6= i. If it is possible to pick U from some prefixed family S ⊂ τ, we say that the family is ∞ isolated with respect to S. If there is a decomposition I = n=1 In such that every family {Hi : i ∈ In} is isolated (with respect to S), then the family {Hi : i ∈ I} is said to be σ-isolated (with respect to S). Finally a family N of subsetsS of X is said to be a network for the topology of X if every open set is a union of members of N. Definition 1.1. A compact Hausdorff space K is said to be a descriptive compact space if its topology has a σ-isolated network. The class of topological spaces having a σ-isolated network generalizes in a natural way metrizable spaces (Bing-Nagata-Smirnov Theorem, see e.g. [15]). These spaces were first studied by Hansell in its pioneering work [10] recently published in [12]. He proved there, among other results, that if a topological space is fragmented by a finer metric, then it has σ-isolated network if, and only if, it has a certain covering property, namely, the space is hereditarily weakly θ-refinable. Descriptive Banach spaces have been studied in [12, 20] and also in the context of renorming theory in [17, 18, 22, 23, 24]. The first and more clear example of a descriptive compact space one may come across is any metrizable compact. Embeddings into c0(Γ) of Eberlein compacta show that they are descriptive, since (c0(Γ), pointwise) has a σ-isolated network, [12, 20]. More generally, Gul’ko compact spaces are shown to be descriptive too, see for instance [25]. Scattered compacta K with K(ω1) = ∅ are also descriptive: just consider as singletons the points (α) (α+1) of each relatively discrete set {K \ K : 0 ≤ α < γ} where γ < ω1 is such that K(γ) = ∅. We shall show later, Lemma 4.6, that Corson compact spaces defined by “al- most disjoint families of sets” are also descriptive, which includes an interesting example of compactum built by Argyros and Mercourakis [1]. On the other hand, the compactum Date: 16th May 2007. 2000 Mathematics Subject Classification. 46B26, 46B03, 46B20, 54D20. Key words and phrases. weak∗ locally uniformly rotund; renorming; descriptive compact spaces; fragmentability. Research partially supported by the D.G.E.S. grant PB98-0381. 1 2 [0, ω1] is not descriptive [12], see also Example 4.4. Let us turn now our attention to renorming problems. Some results have been obtained recently showing that geometrical properties such as the existence of equivalent Kadec or locally uniformly rotund (LUR) norms in a Banach space X can be characterized by the existence of certain types of networks of the norm topology which are σ-isolated for the weak topology of X (LUR norms [17, 18, 23], dual LUR norms [23, 24] and Kadec norms [22]). Recently, in [8], it has been proved that the dual unit ball (with its weak∗ topology) is uniformly Eberlein if, and only if, the dual space has a w∗-UR equivalent norm, which is equivalent to X have a uniformly Gateaux smooth equivalent norm. We shall introduce in general the concept of τ-LUR norm, for τ a locally convex topology. Definition 1.2. A norm k.k on X is said to be τ-locally uniformly rotund at some point x ∈ X if for every (xn) ⊂ X with limn kxnk = kxk and limn kx + xnk = 2kxk, then τ-limn xn = x. A norm k.k on X is said to be τ-locally uniformly rotund (τ-LUR) if it is τ-lower semi- continuous and τ-locally uniformly rotund at x for every x ∈ X. For the case of the weak topology, Molto,´ Orihuela, Troyanski and Valdivia [18] proved that a w-LUR Banach space has an equivalent LUR norm. Mercourakis showed that the ′ space c1(Σ × Γ) has a pointwise-LUR equivalent norm. He used that fact to build an equivalent w∗-LUR norm in a dual of a WCD Banach space, see also [6]. The main result in [25] shows that a dual Banach space X∗ admits an equivalent w∗-LUR norm if, and ∗ only if, (BX∗ , w ) is a descriptive compact space. In this paper we give sufficient and necessary conditions on a Banach space X and a locally convex topology τ to obtain a τ-LUR norm (equivalent or coarser) on X. For ∗ ∗ F ⊂ X a total subspace we shall consider its associated norm pF (x) = sup{x (x) : ∗ x ∈ BX∗ ∩ F }. Recall that F is said norming if pF is an equivalent norm on X. Our main result is about “descriptively generated spaces”: Theorem 1.3. Let X be a Banach space, F ⊂ X∗ a total subspace and K ⊂ X a descrip- tive σ(X,F )-compact subset such that X = spank.k(K). Then the following affirmations hold: (1) X admits a coarser σ(X,F )-LUR norm and the topology σ(X,F ) on X has a σ-isolated network. (2) Moreover, if K is fragmented by pF , then X admits a coarser norm which is pF - LUR and σ(X,F )-lower semi-continuous. (3) The norms given in (1) and (2) can be taken equivalent to the original norm of X if, and only if, F is norming. This theorem covers both the cases of weakly compactly generated Banach spaces and ∗ dual Banach spaces such that (BX∗ , w ) is a descriptive compactum. Since a weakly com- pact subset of a Banach space is descriptive and norm fragmented, we get Troyanski’s result: a WCG Banach space is LUR renormable, see [6]. The theorem also applies to Banach spaces with a Markusevich basis. Indeed, if {xα, fα} is a M-basis on X, and k.k F = span {fα}, then ({xα} ∪ {0}) is a descriptive σ(X,F )-compact that generates X. Let us say that statement (2) above also can be deduced using results from [24]. The topological properties of Banach spaces generated by a norm fragmentable compact space has been recently studied in [5]. Recall that descriptive compact spaces are fragmentable [25], see Corollary 2.6 for a self-contained proof. In section 2 we study the structure of a descriptive compactum with respect to a finer fragmenting metric. Section 3 is devoted to renorming, including the proof of Theorem 1.3. In the last section we show that the class of descriptive compacta has a 3 behaviour similar to the class of fragmentable compacta, studied by Ribarska [26], see also [7]. Moreover, descriptive compact spaces have nicer properties than fragmentable ones, see Proposition 4.2. We also discuss some examples to give an account of how wide is the class of descriptive compacta. Part of the results of this paper appeared in a preliminary version of [25]. 2. SPACES WITH σ-ISOLATED NETWORK Hansell’s definition of descriptive topological spaces [12], later called isolated-analytic spaces in [11], is quite technical. In the case of a compact topological space, being de- scriptive in the sense of Hansell is equivalent to say that it satisfies Definition 1.1. The following definition has been used in [22, 23, 24]. Definition 2.1. Let S1 and S2 be families of subsets of a given set X. We say that X has P (S1, S2) with a sequence (An) of subsets of X if for every x ∈ X and every V ∈ S1 with x ∈ V , there is n ∈ N and U ∈ S2 such that x ∈ An ∩ U ⊂ V . One can easily realize that this generalized property P is also transitive, that is, if X has P (S1, S2) and P (S2, S3) then X has P (S1, S3). The following result links our property P with the existence of σ-isolated networks. Implication i) ⇔ iii) appears in [10] for topologies. Theorem 2.2. Let (X,τ) be a regular topological space and let S a subfamily of τ. The following statements are equivalent: i) τ has a network which is σ-isolated with respect to S. ii) There is a finer metric d such that X has P (d, S). iii) There is a finer metric d such that X has P (d, S) with τ-closed sets. iv) There exists a finer metric d, τ-closed sets An and families {Ui : i ∈ In} of τ- open sets which are union of sets from S, such that the families {An ∩Ui : i ∈ In} are disjoint and {An ∩ Ui : n ∈ N, i ∈ In} is a network for d.
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