Commutativity preserving linear maps on central simple algebras.
type: Item
instance of: scholarly article
title: Commutativity preserving linear maps on central simple algebras.
zbMATH Open document ID: 1072.16028
author: Matej Brešar
author: Peter Šemrl
published in: Journal of Algebra
publication date: 2005-02-22T00:00:00Z
review text: Let \(A\) be a unital algebra over a field \(F\) and \(\varphi\colon A\to A\) a linear map. Then \(\varphi\) is called commutativity-preserving if for all \(x,y\in A\) such that \(xy=yx\), \(\varphi(x)\varphi(y)=\varphi(y)\varphi(x)\). Among the obvious examples are the standard commutativity-preserving maps, defined by \(\varphi(x)=\lambda\psi(x)+\mu(x)1\), where \(\lambda\in F\), \(\psi\) is an automorphism or antiautomorphism of \(A\), and \(\mu\) is a linear functional on \(A\). In this paper it is shown that if \(\text{char}(F)=0\) and \(A\) is a central simple algebra with \(4\neq\dim_FA<\infty\), then a linear map \(\varphi\) such that \(\varphi(x^2)\varphi(x)=\varphi(x)\varphi(x^2)\) for all \(x\in A\) is either a standard commutativity-preserving map or a map with commutative range. This result extends a recent result of \textit{M. Omladič}, \textit{H. Radjavi} and \textit{P. Šemrl} [J. Pure Appl. Algebra 156, No. 2-3, 309-328 (2001; Zbl 0973.15002)], which dealt with the case of an algebraically closed \(F\) of characteristic \(0\) and \(A=M_n(F)\), \(n\neq 2\).
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 15A04
zbMATH DE Number: 2136966
zbMATH Keywords: central simple algebras
zbMATH Keywords: commutativity-preserving maps
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.09.014
OpenAlex ID: W2029771737
cites work: Q4780925
cites work: Centralizers in Free Associative Algebras
cites work: Commuting Traces of Biadditive Mappings, Commutativity-Preserving Mappings and Lie Mappings
cites work: Commuting maps: a survey.
cites work: Preserving commutativity
cites work: Linear maps that preserve commuting pairs of matrices
DOI: 10.1016/J.JALGEBRA.2004.09.014
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description: scientific article; zbMATH DE number 2136966