Orthogonality preserving linear maps on \(C^{\ast }\)-algebras with non-zero socles
type: Item
instance of: scholarly article
title: Orthogonality preserving linear maps on \(C^{\ast }\)-algebras with non-zero socles
zbMATH Open document ID: 1279.47059
author: Yanyan Li
published in: Journal of Mathematical Analysis and Applications
publication date: 2013-04-22T00:00:00Z
review text: Let \(\mathcal{A}\) be a \(C^{\ast }\)-algebra and \(a,b\in \mathcal{A}\). If \(ab^{\ast }=b^{\ast }a=0,\) we write \(a\bot b\) and say that \(a\) and \(b\) are orthogonal. A linear map \(T:\mathcal{A\rightarrow B}\) between \(C^{\ast }\)-algebras \(\mathcal{A}\) and \(\mathcal{B}\) is said to be orthogonality preserving if \(a\bot b\) implies \(T(a)\bot T(b)\) for every \(a,b\in \mathcal{A}\). We say that \(T\) is biorthogonality preserving if \(a\bot b\) if and only if \(T(a)\bot T(b)\) for every \(a,b\in \mathcal{A}\). The paper has three chapters. In the first chapter, the author introduces notations and presents known results while in the second chapter some auxiliary results are proved. The first of the two main results which are presented and proved in the third chapter is the following theorem. Let \(\mathcal{A}\) and \(\mathcal{B}\) be \(C^{\ast }\)-algebras and denote by Soc\((\mathcal{A})\) the socle of \(\mathcal{A}\). Suppose that \(\mathcal{A}\) is unital with unit \(1\) and with non-zero socle. Let \(T:\mathcal{A\rightarrow B}\) be an orthogonality preserving bijective linear map with \(\{T(1)\}^{\bot}=\{0\}\), where \(\{T(1)\}^{\bot }=\{y\in \mathcal{B}:y\bot T(1)\}.\) Then \[ T^{-1}(T(1)^{\ast }T(a)-T(a^{\ast })^{\ast }T(1))\text{Soc}(\mathcal{A})=\{0\}, \] and \[ T^{-1}(T(a^{2})T(1)^{\ast }T(1)-T(a)T(a^{\ast })^{\ast }T(1))\text{Soc}(\mathcal{A})=\{0\} \] for every \(a\in \mathcal{A}\). Moreover, if Soc\((\mathcal{A})\) is essential, then \(\mathcal{B}\) is unital, and \(T\) is a Jordan *-isomorphism multiplied by an invertible element. As a direct consequence of this theorem, it follows that every biorthogonality preserving surjective linear map from a unital \(C^{\ast }\)-algebra with essential socle to a \(C^{\ast }\)-algebra is automatically bounded and a multiple of a Jordan *-isomorphism. In conclusion, the author studies orthogonality preserving bijective linear maps \(T\) from a unital \(C^{\ast }\)-algebra with a non-zero socle to a \(C^{\…
reviewed by: Janko Marovt
Mathematics Subject Classification ID: 47B49
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L40
Mathematics Subject Classification ID: 17C65
zbMATH DE Number: 6156257
zbMATH Keywords: linear preservers
zbMATH Keywords: \(\mathbb{C}^\ast\)-algebras
zbMATH Keywords: socle
zbMATH Keywords: Jordan homomorphism
MaRDI profile type: Publication
full work available at URL: j.jmaa.2012.11.026
OpenAlex ID: W2042091417
DOI: 10.1016/J.JMAA.2012.11.026
description: scientific article; zbMATH DE number 6156257