On maps preserving products of matrices
type: Item
instance of: scholarly article
title: On maps preserving products of matrices
zbMATH Open document ID: 1430.15021
author: Louisa Catalano
author: Samuel Hsu
author: Regan Kapalko
published in: Linear Algebra and its Applications
publication date: 2019-01-24T00:00:00Z
review text: Let \(R=M_n(D)\), where \(n\ge 2\) and \(D\) is a division ring with characteristic different from \(2\), let \(Z\) be the center of \(R\), and let \(m\) and \(k\) be (fixed) invertible elements in \(R\). It is shown that if \(f:R\to R\) is a bijective additive map satisfying \(f(x)f(y) = m\) whenever \(x,y\in R\) are such that \( xy = k\), then there exists an automorphism or an antiautomorphism \(\varphi\) of \(R\) such that \(f(x) = f(1)\varphi(x)\) for all \(x \in R\). In the case where \(D=\mathbb C\), the authors also consider a similar condition in which the role of the ordinary product is replaced by the Jordan product \(x\circ y=xy+yx\).
reviewed by: Matej Brešar
Mathematics Subject Classification ID: 15A86
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16R60
zbMATH DE Number: 7006576
zbMATH Keywords: maps preserving products
zbMATH Keywords: Jordan product
zbMATH Keywords: matrices over division rings
MaRDI profile type: Publication
full work available at URL: j.laa.2018.10.029
OpenAlex ID: W2899349492
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DOI: 10.1016/J.LAA.2018.10.029
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description: scientific article; zbMATH DE number 7006576