Renormings and fixed point property in non-commutative \(L_1\)-spaces. II: Affine mappings
type: Item
instance of: scholarly article
title: Renormings and fixed point property in non-commutative \(L_1\)-spaces. II: Affine mappings
zbMATH Open document ID: 1260.46004
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2012-07-16T00:00:00Z
reviewed by: B. Turett
Mathematics Subject Classification ID: 46B03
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 6057313
zbMATH Keywords: fixed point property
zbMATH Keywords: affine mappings, nonexpansive mappings
zbMATH Keywords: von Neumann algebras
zbMATH Keywords: noncommutative \(L_1\)-spaces
zbMATH Keywords: renorming
author: Carlos A. Hernández-Linares
author: Maria A. Japón
MaRDI profile type: Publication
full work available at URL: j.na.2012.04.050
OpenAlex ID: W1985508622
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DOI: 10.1016/J.NA.2012.04.050
review text: A Banach space has the fixed point property for affine mappings (A-FPP) if every affine nonexpansive mapping from a closed bounded convex subset of \(X\) into itself has a fixed point. The authors note that, since Banach spaces containing asymptotically isometric copies of \(\ell_1\) fail to have the A-FPP, every nonreflexive subspace of \(L_1[0,1]\) or, more generally, every nonreflexive subspace of a noncommutative \(L_1\)-space associated with a finite von Neumann algebra with its usual norm fails to have the A-FPP. The main result in the article is that, if \(\mathcal{M}\) is a finite von Neumann algebra, then the noncommutative \(L_1\)-space \(L_1(\mathcal{M})\) can be renormed to have the A-FPP. In particular, \(L_1[0,1]\) can be endowed with a norm \(|\!|\!|\cdot |\!|\!|\) equivalent to \(\|\cdot\|_1\) so that \((L_1[0,1], |\!|\!|\cdot |\!|\!|)\) has the A-FPP. Whether \(L_1[0,1]\) can be renormed to have the fixed point property for nonexpansive mappings is still an open question.NEWLINENEWLINE For Part I see [ibid. 74, No. 10, 3091--3098 (2011; Zbl 1230.46010)].
description: scientific article; zbMATH DE number 6057313