The convex point of continuity property in Banach spaces not containing \(\ell _{1}\)
type: Item
instance of: scholarly article
title: The convex point of continuity property in Banach spaces not containing \(\ell _{1}\)
zbMATH Open document ID: 1221.46010
author: José A. Soler-Arias
author: Ginés López-Pérez
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-03-29T00:00:00Z
review text: Recall that a closed, bounded and convex subset \(C\) of a Banach space is said to have the point of continuity property (PCP) (resp., the convex point of continuity property (CPCP), provided that every nonempty closed and bounded subset of \(C\) (resp., every nonempty closed, bounded and convex subset) admits a point of continuity of the identity map from the weak to norm topologies. A Banach space \(X\) satisfies PCP (resp., CPCP) if its closed unit ball has PCP (resp. CPCP). A well-known open problem is if PCP and CPCP are determined by the separable subspaces. The main goal of the paper under review is to prove that, for Banach spaces not containing subspaces isomorphic to \(l_{1}\), CPCP is determined on subspaces with a Schauder basis. In order to prove this, first the authors obtain a local characterization of CPCP for closed, bounded and convex subsets in Banach spaces without \(l_{1}\)-copies. This characterization is an extension of a result in [\textit{S. Argyros, E. Odell} and \textit{H. Rosenthal}, Lect. Notes Math. 1332, 80--111 (1988; Zbl 0658.46012)].
Mathematics Subject Classification ID: 46B03
Mathematics Subject Classification ID: 46B22
zbMATH DE Number: 5871229
zbMATH Keywords: convex point of continuity property
zbMATH Keywords: Banach spaces not containing \(l_{1}\)
reviewed by: Elói M. Galego
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.01.052
OpenAlex ID: W2060927333
cites work: Q3806166
cites work: Q3857296
cites work: Dentability and finite-dimensional decompositions
cites work: Martingales valued in certain subspaces of \(L^1\)
cites work: \(G_{\delta}\)-embeddings in Hilbert space. II
cites work: A Banach Space Not Containing c 0 , l 1 or a Reflexive Subspace
cites work: An extension of a structure theorem of Bourgain
cites work: Banach spaces whose duals are \(L_ 1\) spaces and their representing matrices
cites work: The Poulsen simplex
cites work: Q4136031
cites work: The convex point of continuity property in Asplund spaces
cites work: Some remarks on \(\ell^1\)-sequences
cites work: On separable Lindenstrauss spaces
cites work: A simplex with dense extreme points
cites work: Banach Spaces With Separable Duals
DOI: 10.1016/J.JMAA.2011.01.052
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description: scientific article; zbMATH DE number 5871229