An extension of the Gleason-Kahane-Żelazko theorem: A possible approach to Kaplansky's problem
type: Item
instance of: scholarly article
title: An extension of the Gleason-Kahane-Żelazko theorem: A possible approach to Kaplansky's problem
zbMATH Open document ID: 1154.47030
author: Matej Brešar
author: Peter Šemrl
published in: Expositiones Mathematicae
publication date: 2008-08-06T00:00:00Z
review text: Let \(A\) and \(B\) be two complex unital Banach algebras with \(B\) semisimple. In this nice survey, the authors are concerned with Kaplanski's problem that asks under which conditions every unital, invertibility preserving linear map \(\phi: A\to B\) is a Jordan homomorphism. Note that a linear map \(\phi:A\to B\) is a Jordan homomorphism if \(\phi(x^2)= \phi(x)^2 \) for all \(x\in A\). It is proved that if, additionally, \(\phi\) is a surjection that locally preserves commutativity -- meaning that \([\phi(x^2),\phi(x)]=0\), where \([x,y]=xy-yx\) is the Lie product --, then the answer is positive. As a corollary, they obtain a new proof of the Marcus--Purves theorem [\textit{M.\,Marcus} and \textit{R.\,Purves}, Can.\ J.\ Math.\ 11, 383--396 (1959; Zbl 0086.01704)] that any unital linear map \(\phi: M_n\to M_n\) preserving invertibility is a Jordan homomorphism. Here, as usual, \(M_n\) is the algebra of all \(n\times n\) complex matrices.
reviewed by: Raymond Mortini
Mathematics Subject Classification ID: 47B48
Mathematics Subject Classification ID: 46J99
zbMATH DE Number: 5307074
zbMATH Keywords: Jordan homomorphisms
zbMATH Keywords: inverrtibility preserving linear maps
zbMATH Keywords: semi-simple Banach algebras
zbMATH Keywords: linear preserver
zbMATH Keywords: commutativity
MaRDI profile type: Publication
full work available at URL: j.exmath.2007.11.004
OpenAlex ID: W2055995410
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DOI: 10.1016/J.EXMATH.2007.11.004
description: scientific article; zbMATH DE number 5307074