A non-reflexive Grothendieck space that does not containl ∞

Haydon, Richard

doi:10.1007/bf02761818

Cited by 96 publications

(68 citation statements)

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“…We will consider several separation properties in Boolean algebras. This is a natural generalization of the following well-known concept ( [5]). …”

Section: Independent Familiesmentioning

confidence: 99%

“…The subsequential completeness property was introduced in [5]. That paper included applications of the subsequential completeness property in the theory of Banach spaces as well as a result of Argyros (Proposition 1G) that every Boolean algebra with the subsequential completeness property has an uncountable independent family.…”

Section: Definition 13mentioning

confidence: 99%

“…It follows that many Banach spaces present in the literature have l ∞ as a quotient, in particular the spaces of [5] or [9]. In [9], besides Boolean separations a lattice version of the subsequential completeness property is considered for connected compact K. Namely (see [9,5.1]) we consider spaces K such that given any pairwise disjoint (…”

Section: Corollary 15 If a Is A Boolean Algebra Having A Subsequentmentioning

confidence: 99%

“…In [9], besides Boolean separations a lattice version of the subsequential completeness property is considered for connected compact K. Namely (see [9,5.1]) we consider spaces K such that given any pairwise disjoint (…”

Section: Corollary 15 If a Is A Boolean Algebra Having A Subsequentmentioning

confidence: 99%

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Independent families in Boolean algebras with some separation properties

Koszmider

Shelah

2013

Algebra Univers.

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Abstract. We prove that any Boolean algebra with the subsequential completeness property contains an independent family of size c, the size of the continuum. This improves a result of Argyros from the 1980s, which asserted the existence of an uncountable independent family. In fact, we prove it for a bigger class of Boolean algebras satisfying much weaker properties. It follows that the Stone space K A of all such Boolean algebras A contains a copy of theČech-Stone compactification of the integers βN and the Banach space C(K A ) has l∞ as a quotient. Connections with the Grothendieck property in Banach spaces are discussed.

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“…We will consider several separation properties in Boolean algebras. This is a natural generalization of the following well-known concept ( [5]). …”

Section: Independent Familiesmentioning

confidence: 99%

Section: Definition 13mentioning

confidence: 99%

Section: Corollary 15 If a Is A Boolean Algebra Having A Subsequentmentioning

confidence: 99%

Section: Corollary 15 If a Is A Boolean Algebra Having A Subsequentmentioning

confidence: 99%

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Independent families in Boolean algebras with some separation properties

Koszmider

Shelah

2013

Algebra Univers.

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“…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.…”

Section: I) (X Weak) Is Sigma-fragmented By a Metric Which Is Strongmentioning

confidence: 99%

A note on nonfragmentability of Banach spaces

Mirmostafaee

2001

International Journal of Mathematics and Mathematical Sciences

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Abstract. 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 RadonNikodym 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

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Limited Sets in C(K)‐Spaces and Examples Concerning the Gelfand‐Phillips‐Property

Schlumprecht

1992

Mathematische Nachrichten

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In this paper we give criteria for limitedness in C(K)-spaces and discuss the GELFAND-PHILLIPS-property. We show that the GELFAND-PHILLIPS-property is not a three-space-property, that I , $t x does not imply the GELFAND-PHILLiPS-prOperty of x and that the GELFAND-PHILLIPS-prOperty of a space X does not imply that the dual unit ball contains a w*-sequentially precompact subset which norms X up to a constant.

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A non-reflexive Grothendieck space that does not containl ∞

Cited by 96 publications

References 5 publications

Independent families in Boolean algebras with some separation properties

Independent families in Boolean algebras with some separation properties

A note on nonfragmentability of Banach spaces

Limited Sets in C(K)‐Spaces and Examples Concerning the Gelfand‐Phillips‐Property

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