On a problem of Rolewicz about Banach spaces that admit support sets
type: Item
instance of: scholarly article
title: On a problem of Rolewicz about Banach spaces that admit support sets
zbMATH Open document ID: 1187.46011
author: Piotr Koszmider
published in: Journal of Functional Analysis
publication date: 2009-10-20T00:00:00Z
review text: A nonempty closed convex set \(C\) in a Banach space \(X\) is called a \textit{support set} if for every \(c\in C\) there exists \(x^*\in X^*\) such that \(x^*(c)=\inf\{x^*(x):\;x\in C\}<\sup\{x^*(x):\;x\in C\}\). In [\textit{S.\,Rolewicz}, ``On convex sets containing only points of support'', Commentat.\ Math., Spec.\ Vol.\,I, dedic.\ L.\,Orlicz, 279--281 (1978; Zbl 0385.46009)], it was proved that there are no separable support sets, and it was asked whether every nonseparable Banach space contains a support set. In the present paper, it is shown that this is not the case, by providing an example of a nonseparable Banach space (consistent with the axioms of ZFC) of type \(C(K)\) that contains no support set. It was proved in [\textit{S.\,Todorčević}, ``Biorthogonal systems and quotient spaces via Baire category methods'', Math.\ Ann.\ 335, No.\,3, 687--715 (2006; Zbl 1112.46015)] that it is consistent that any nonseparable Banach space contains an uncountable biorthogonal system. It follows that the nonexistence of the example in the present paper is also consistent with the axioms of ZFC, so Rolewicz's problem turns out to be undecidable. The example provided here is, as mentioned, a space of type \(C(K)\), and it is hereditarily Lindelöf in the weak topology. It does not have any nonseparable subspace nor any nonseparable quotient of type \(C(M)\) for \(M\) dispersed compact, so, unlike Kunen's space, it is not Asplund. Indeed, every nonseparable \(C(K)\) space for \(K\) dispersed admits a support set, as follows from results in [\textit{A.\,J.\thinspace Lazar}, ``Points of support for closed convex sets'', Ill.\ J.\ Math.\ 25, 302--305 (1981; Zbl 0437.46008)]. The construction of the example requires the study of the properties of the Boolean algebra of clopen sets of the compact set \(K\) (\(K\) is totally disconnected, so it is the Stone space of a Boolean algebra; the space \(K\) is obtained somehow from the Cantor set by a splitting procedure). This is done…
reviewed by: V. Montesinos
Mathematics Subject Classification ID: 46B26
Mathematics Subject Classification ID: 46B25
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 03E35
zbMATH DE Number: 5618753
zbMATH Keywords: Banach spaces
zbMATH Keywords: biorthogonal systems
zbMATH Keywords: Banach spaces of continuous functions
zbMATH Keywords: support sets
zbMATH Keywords: independence results
MaRDI profile type: Publication
full work available at URL: j.jfa.2009.07.016
OpenAlex ID: W2091555234
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DOI: 10.1016/J.JFA.2009.07.016
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description: scientific article; zbMATH DE number 5618753