Boundaries of Asplund spaces
type: Item
instance of: scholarly article
title: Boundaries of Asplund spaces
zbMATH Open document ID: 1203.46010
published in: Journal of Functional Analysis
publication date: 2010-09-06T00:00:00Z
review text: Let \(K\) be a \(w^*\)-compact convex subset of a dual Banach space \(X^*\). A subset \(B\subset K\) is called a (James) \textit{boundary} of \(K\) if, for any \(x\in X\), there is \(f\in B\) with \(\max x(K)=f(x)\). A boundary \(B\) is called \textit{strong} if the norm closure of its closed convex \(\overline{\mathrm{co}}B=K\). It is well-known that every (norm) separable boundary is strong and that there are non-strong boundaries. In the paper under review, the authors investigate when a boundary is strong. Let us present two results. Let \(X\) be an Asplund space and \(B\) be a boundary of the unit ball of \(X^*\). Then \(B\) is strong if, and only if \(B\) is FSP. A Banach space \(X\) does not contain \(\ell_1\) if and only if, for every \(w^*\)-compact subset \(K\) of \(X^*\), any \(w^*\)-\(K\)-analytic boundary \(B\) of \(K\) is strong. By the authors' definition, a subset \(C\) of \(X^*\) is \textit{finitely self-predicable} (FSP) if, there is a map \(\xi\) from the family \(\mathcal{F}_X\) of all finite subsets of \(X\) into the family of all finite subsets of \(\mathrm{co}\,C\) such that, for any increasing sequence \(\sigma_n\) in \(\mathcal{F}_X\) with \(E=\overline{\mathrm{lin}}(\sigma_n)_1^{\infty}\) and \(D=\bigcup_n\xi(\sigma_n)\), the restriction \(C|_E\) belongs to \(\overline{\mathrm{co}}\,D|_E\). The authors also study the relationship between the classical combinatorial inequality of Simons for boundaries and the more recent (I)-property of Fonf and Lindenstrauss.
reviewed by: A. M. Plichko
Mathematics Subject Classification ID: 46B26
Mathematics Subject Classification ID: 46B03
Mathematics Subject Classification ID: 46B10
zbMATH DE Number: 5779336
zbMATH Keywords: Banach spaces
zbMATH Keywords: Asplund spaces
zbMATH Keywords: James boundary
zbMATH Keywords: inequality of Simons
zbMATH Keywords: (I)-property
author: B. Cascales
author: V. P. Fonf
author: José Orihuela
author: S. Troyanski
MaRDI profile type: Publication
full work available at URL: j.jfa.2010.04.010
OpenAlex ID: W2028514227
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DOI: 10.1016/J.JFA.2010.04.010
description: scientific article; zbMATH DE number 5779336