A NOTE ON NONFRAGMENTABILITY OF BANACH SPACES

We use Kenderov-Moors characterization of fragmentability to show that if a compact Hausdorff space X with the tree-completeness property contains a disjoint sequences of clopen sets, then (C(X), weak) is not fragmented by any metric which is stronger than weak topology. In particular, C(X) does not admit any equivalent locally uniformly convex renorming. 2000 Mathematics Subject Classification. 46B20, 46B26, 46E15, 54C45.

1. Introduction.Let (X, τ) be a topological space and ρ a metric on X.Given > 0, a nonempty subset A of X is said to be fragmented by ρ down to if each nonempty subset of A has a nonempty τ-relatively open subset of A with ρ-diameter less than .The set A is said to be fragmented by ρ if A is fragmented by ρ down to for each > 0. The set A is said to be sigma-fragmented by ρ [7] if for each > 0, A can be expressed as A = ∞ n=1 A n, with each A n, fragmented by ρ down to .The notion of fragmentability was originally introduced in [11] as an abstraction of phenomena often encountered, for example, in Banach spaces with the Radon-Nikodym property, in weakly compact subsets of Banach spaces and in the dual of Banach spaces.The notion of σ -fragmentability appeared in [10] in order to extend the study of compact fragmented space to noncompact spaces.It turns out that the question of whether a given Banach space with weak topology is sigma-fragmented by the norm is closely connected with the question of the existence of an equivalent Kadec and locally uniformly convex norm.The reader may refer to [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20] for some application of fragmentability and its variants in other topics of Banach spaces.
Kenderov and Moors [13,14] used the following topological game to characterize fragmentability and sigma-fragmentability of a topological space X.
Two players Σ and Ω alternatively select subsets of X.The player Σ usually starts the game by choosing some nonempty subset A 1 of X, then the Ω-player chooses some nonempty relatively open subset A 1 , say B 1 , then Σ will choose a nonempty set A 2 ⊂ B 1 and in turn, Ω picks up some nonempty relatively open subset B 2 of A 2 .By continuing this procedure, the two players generate a sequence of sets which is called a play and is denoted by p = (A i ,B i ) ∞ i=1 .If are the first "n" move of some play (of the game), then we call p k a partial play of the game.The player Ω is said to have won the play if ∞ i=1 A i contains at most one point.Otherwise the player Σ is said to have won this play.Under the term strategy for the player Ω, we understand a mapping ω which assigns to every partial play p n a nonempty relatively open subset i=1 is called an ω-play if B i = ω(p i ) for every i ≥ 1.Similarly, the partial play p n is called a partial ω-play, if B i = ω(p i ) for each i < n.The map ω is called a winning strategy for the player Ω if he/she wins every ω-play.If the space X is fragmentable by a metric d(•, •), then Ω has an obvious winning strategy ω.Indeed, to each partial play p n this strategy puts into correspondence some nonempty subset B n ⊂ A n which is relatively open in A n and has d-diameter less than 1/n.Clearly, the set i≥1 A i = i≥1 B i has at most one point because it has d-diameter 0. It turns out that the existence of a winning strategy for player Ω characterizes fragmentability.
Theorem 1.1 (see [13]).The topological space X is fragmentable if and only if the player Ω has a winning strategy.
By Theorem 1.1, it was shown in [15] that X/c 0 , where X is the Haydon-Zizler subspace of ∞ [5] is not fragmented by any metric.According to a result of Ribarska [18], if a Banach space admits an equivalent strictly convex renorming, then it is fragmented by a metric.It follows that X/c 0 does not admit strictly convex renorming.This could be considered as an extension of [1].
Although ∞ taken with its weak topology is not sigma-fragmented by the norm, it is fragmented by a lower semi-continuous metric (see [9,Example 3.2]).However, in [14], it is shown that fragmentability and sigma-fragmentability in a Banach space may be related to each other in the following way.Theorem 1.2 (see [14, Theorems 1.3, 1.4, and 2.1]).For a Banach space X the following are equivalent: (i) (X, weak) is sigma-fragmented by a metric which is stronger than the weak topology; (ii) (X, weak) is fragmented by a metric which is stronger than the weak topology; (iii) there exists a strategy ω for the player Ω such that, for every ω-play p = It is known that whenever X is compact and extremely disconnected, then C(X) contains an isometric copy of ∞ (see [2, page 18]), therefore it is not sigma-fragmented by the norm.However, there exists a compact Hausdorff space X (with the tree completeness property) such that C(X) does not contain a copy of ∞ (see [4]).It is natural to ask if such a space is sigma-fragmented by the norm.The above result enable us to give an answer to this question.More precisely, thanks to Theorem 1.2, we will show that if a compact Hausdorff space X with the tree-completeness property has a sequence of disjoint clopen sets, then (C(X), weak) is not (sigma) fragmented by any metric which is stronger than the weak topology.It follows that C(X) does not admit any equivalent locally uniformly convex norm.

Results
The elements of T , are finite (possibly empty) strings of 0's and 1's.The empty string ( ) is the unique string of length 0; more generally, the length |t| of a string t is n if t ∈ {0, 1} n .The tree-order is defined by s ≺ t if |s| < |t| and t(m) = s(m) for m ≤ |s|.Each t ∈ T has exactly two immediate successors, that is, t0 and t1.
A topological space X is said to have the tree-completeness property if whenever {V t } t∈T is a sequence of disjoint clopen sets in X there exists some b ∈ {0, 1} N * , N * = N ∪ {0}, such that n∈N * V b|n is open.Evidently, every infinite extremally disconnected space [3] has the tree-completeness property.However, as it was mentioned in Section 1, there exists a compact Hausdorff space with the tree-completeness property which is not extremally disconnected.
for i = 0, 1 and t ∈ T .Then {Z t } t∈T is a sequence of disjoint clopen subsets of X.By the tree-completeness property of X, there exists some b ∈ {0, 1} N * , such that is clopen in X, thus it is C * -embedded.
Lemma 2.3.Let {V n } n∈N be an infinite disjoint sequence of clopen subsets of a compact Hausdorff space X and µ ∈ C(X) * , where X has the tree-completeness property.Then there exists an infinite set Proof.Suppose that 2 µ < n .Note that for every infinite subset M of N, there exists some infinite subset If the lemma were not true, we can find infinite disjoint subsets M 1 ,...,M n of N and continuous functions f 1 ,...,f n such that Theorem 2.4.Let X be a compact Hausdorff space with the tree-completeness property.If X contains a disjoint sequence of clopen sets.Then (C(X), weak) is not (sigma) fragmented by any metric which is stronger than weak topology.
Proof.By Theorem 1.2, it is enough to show that for each strategy ω for the player Ω there exists an ω-play p = (A i ,B i ) i such that, i≥1 B i ≠ ∅ and lim i→∞ normdiam(B i ) > 0. Fix a strategy ω for the player Ω.By induction on |t|, t ∈ T , we will construct partial ω-plays p t = (A ( ) ,B ( ) ,A t|1 ,...,A t ).Then, we will show that there is some b ∈ {0, 1} N * , such that the ω-play p b = (A ( ) ,B ( ) ,A b|1 ,...) has the required properties.
Let {V n } n∈N be an infinite disjoint sequence of nonempty clopen subsets of X.Let N ( ) be an infinite subset of N such n∈N ( ) V n is a clopen subset of X.For some f ( ) in the unit ball of C(X), we define as the first choice of the player Σ.
Applying Lemma 2.3, we can find an infinite subset N t of N t , such that n∈N t V n is clopen and Suppose that N t 0 and N t 1 are two disjoint infinite subset of N t , such that each n∈N ti V n is clopen, i = 0, 1.Let f ti = f t .χX\ n∈N ti Vn and define (ii) for each N t , n∈N t V n is clopen, (iii) N t ⊂ N s , when s ≺ t, (iv) N t ∩ N s = ∅, when s and t are not comparable, (v) norm-diam(A t ) = 1 for each t ∈ T , (vi) f t (x) = f ti (x) for x ∈ X \ n∈N t V n and i = 0, 1. Applying Lemma 2.2, we can find some b ∈ {0, 1} N * , such that every continuous function on n∈N * (X \ k∈N b|n V k ) has a continuous extension on X.By (vi), the function f * b (x) = lim n→∞ f b|n (x) is continuous on n∈N * (X \ k∈N b|n V k ), thus it has a continuous extension f b on X without increasing norm.Clearly f b ∈ n∈N * A b|n .Thus A b|n ≠ ∅ and lim n→∞ norm-diam(A b|n ) = 1, that is, the ω-play p b = (A ( ) ,B ( ) ,A b|1 , B b|1 ,...) does not satisfy Theorem 1.2(iii).This proves the theorem.
Corollary 2.5.If a compact Hausdorff space X with the tree-completeness property has an infinite sequence of clopen sets, then C(X) does not admit any equivalent locally uniformly convex norm.
Proof.It is known that if (C(X), weak) admits an equivalent locally uniformly convex norm then it is norm-fragmented (see [7,Theorem 4.2]).Thus the result follows from Theorem 2.4.
every function in C(Y ) can be extended to a function in C(X).Let {N t } t∈T be a sequence of infinite subsets of N, such that (i) N t ⊂ N s , whenever s ≺ t. (ii) N t ∩ N s = ∅, if t and s are not comparable.Let {V n } n∈N * be a sequence of clopen subsets of a compact Hausdorff space X, such that k∈N t V k is open for each t ∈ T .If X has the tree-completeness property, then there exists some Therefore, we have the partial ω-play p ( ) = (A ( ) ), clearly norm-diam(A ( ) ) = 1.Suppose that for every t with |t| ≤ n, the partial ωplay p t = (A ( ) ,B ( ) ,A t|1 ,B t|1 ,...,A t ) has already been defined.Let B t = ω(p t ) be the relatively open subset of A t , chosen by the player Ω according to his/her strategy as the answer to this movement.Let f t ∈ B t , since B t is a relatively open subset of A t , there are linear functionals µ t 1 ,...,µ t K t on C(X) and t > 0, such that