Slices in the unit ball of the symmetric tensor product of \(\mathcal{C}(K)\) and \(L_{1}(\mu )\)
type: Item
instance of: scholarly article
title: Slices in the unit ball of the symmetric tensor product of \(\mathcal{C}(K)\) and \(L_{1}(\mu )\)
zbMATH Open document ID: 1183.46007
author: María D. Acosta
author: Julio Becerra-Guerrero
published in: Arkiv för Matematik
publication date: 2009-09-14T00:00:00Z
review text: The authors consider the symmetric projective tensor product of \(N\) copies of \(C_0(\Omega)\), with \(\Omega\) locally compact, and of \(L_1(\mu)\). The main results are: {\parindent7mm \begin{itemize}\item[(1)] If \(\Omega\) is infinite, then every relatively weakly open subset of the unit ball of \(\bigotimes_{\pi,s}^N C_0(\Omega)\) has diameter 2. \item[(2)] If \(\mu\) is \(\sigma\)-finite and atomless, then every slice of the unit ball of \(\bigotimes_{\pi,s}^N L_1(\mu)\) has diameter 2. \item[(3)] If, in addition, \(N\) is odd, then every relatively weakly open subset of the unit ball of \(\bigotimes_{\pi,s}^N L_1(\mu)\) has diameter 2. \end{itemize}} The results have consequences concerning the roughness of the norm in the dual of \(\bigotimes_{\pi,s}^N X\), the space of \(N\)-homogeneous polynomials on \(X\), for \(X=C_0(\Omega)\) and \(X=L_1(\mu)\).
reviewed by: Dirk Werner
Mathematics Subject Classification ID: 46B04
Mathematics Subject Classification ID: 46B25
Mathematics Subject Classification ID: 46B28
Mathematics Subject Classification ID: 46G25
zbMATH DE Number: 5603926
zbMATH Keywords: diameter-\(2\) property
zbMATH Keywords: symmetric tensor product
zbMATH Keywords: polynomials on Banach spaces
MaRDI profile type: Publication
full work available at URL: s11512-007-0057-6
OpenAlex ID: W122452212
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DOI: 10.1007/S11512-007-0057-6
description: scientific article; zbMATH DE number 5603926