Maps completely preserving commutativity and maps completely preserving Jordan zero-product
type: Item
instance of: scholarly article
title: Maps completely preserving commutativity and maps completely preserving Jordan zero-product
zbMATH Open document ID: 1307.47038
author: Li Huang
author: Yan-Xiao Liu
published in: Linear Algebra and its Applications
publication date: 2014-09-12T00:00:00Z
review text: Let \(X\) and \(Y\) infinite-dimensional Banach spaces over the real or complex field \(\mathbb{F}\), and let \({\mathcal A}\) and \({\mathcal B}\) be standard operator algebras on \(X\) and \(Y\), respectively. A map from \({\mathcal A}\) into \({\mathcal B}\) is said to be \(n\)-commutativity preserving if the induced map from \({\mathcal A}\otimes M_n(\mathbb F)\) into \({\mathcal B}\otimes M_n(\mathbb F)\) preserves the commutativity, and is said to be completely commutativity preserving if it is \(n\)-commutativity preserving map for all \(n\). In this paper, the authors show that every map completely preserving commutativity from \({\mathcal A}\) onto \({\mathcal B}\) is a scalar multiple of either an isomorphism or (in the complex case) a conjugate isomorphism. They also obtain the complete description of all surjective maps between \({\mathcal A}\) and \({\mathcal B}\) that completely preserve the Jordan zero-product of operators.
reviewed by: Abdellatif Bourhim
Mathematics Subject Classification ID: 47B49
Mathematics Subject Classification ID: 47A12
zbMATH DE Number: 6343725
zbMATH Keywords: standard operator algebras
zbMATH Keywords: completely preserver problems
zbMATH Keywords: commutativity
zbMATH Keywords: Jordan zero-product
zbMATH Keywords: isomorphisms
MaRDI profile type: Publication
full work available at URL: j.laa.2014.08.016
OpenAlex ID: W2008675270
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DOI: 10.1016/J.LAA.2014.08.016
description: scientific article; zbMATH DE number 6343725