Relatively weakly open subsets of the unit ball in functions spaces
type: Item
instance of: scholarly article
title: Relatively weakly open subsets of the unit ball in functions spaces
zbMATH Open document ID: 1095.46010
author: Julio Becerra-Guerrero
author: Ginés López-Pérez
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-03-29T00:00:00Z
review text: The existence of a denting point in \(B_X\), the unit ball of a Banach space \(X\), implies the existence of relatively weakly open subsets of \(B_X\) of arbitrary small diameter. In particular, this is the situation in spaces with the RNP. The extreme opposite situation is when all relatively weakly open subsets of \(B_X\) have diameter 2. Let us call such spaces diameter 2 spaces. It is known from \textit{O. Nygaard} and \textit{D. Werner} [Arch. Math. 76, 441--444 (2001; Zbl 0994.46006)] that uniform algebras are diameter 2 spaces. Later on the authors, together with A. Rodriguez, proved that \(C^\ast\)-algebras are diameter 2 spaces as well; cf. \textit{J. Becerra Guerrero, G. Lopez Pérez} and \textit{A. Rodriguez} [J. Lond. Math. Soc. 68, 753--761 (2003; Zbl 1060.46007)]. Let \(K\) be infinite, compact Hausdorff, let \(\tau\) be some locally convex Hausdorff topology on the Banach space \(X\), and denote by \(C(K,(X,\tau))\) the vector space of continuous functions from \(K\) into \((X,\tau)\). By Mackey's theorem, we can give \(C(K,(X,\tau))\) the sup norm when \(\tau\) is compatible with the dual pair \((X,X^\ast)\) or, if \(X\) is a dual, with \((X_\ast,X)\), where \(X_\ast\) is some predual of \(X\). The first result is that in these situations, \(C(K,(X,\tau))\) are diameter 2 spaces. Earlier, Rao showed that there are no denting points in \(B_{C(K,(X,\tau))}\) when \(\tau\) is the weak topology on \(X\), cf. \textit{T. S. S. R. K. Rao} [Proc. Am. Math. Soc. 127, 2969--2973 (1999; Zbl 0932.46013)]. Many corollaries follow; we get the information that often operator ideals are diameter 2 spaces. In particular, is this so for \(L(L_1 (\mu))\), \(L(c_0)\) and \(L(\ell_\infty)\). The second part of the paper is devoted to the study of when operator ideals \(A(L_1(\mu),X)\subset L(L_1(\mu),X)\) are diameter 2 spaces. Here \(\mu\) is a positive measure. The result is that \(A(L_1(\mu),X)\) is not a diameter 2 space if and only if both \(L_1(\mu)\) is finite-dime…
reviewed by: Olav Nygaard
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 54A05
zbMATH DE Number: 5016147
zbMATH Keywords: denting point
zbMATH Keywords: slices
zbMATH Keywords: weakly open subsets
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.06.063
OpenAlex ID: W2030605470
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DOI: 10.1016/J.JMAA.2005.06.063
description: scientific article; zbMATH DE number 5016147