Nonlinear maps preserving Lie products on factor von Neumann algebras
type: Item
instance of: scholarly article
title: Nonlinear maps preserving Lie products on factor von Neumann algebras
zbMATH Open document ID: 1178.47024
author: Fangjuan Zhang
author: Jianhua Zhang
published in: Linear Algebra and its Applications
publication date: 2008-06-09T00:00:00Z
review text: Let \(\mathcal A\) and \(\mathcal B\) be factor von Neumann algebras and \(\Phi : \mathcal A\to \mathcal B\) a bijective map satisfying \(\Phi(AB - BA) = \Phi(A)\Phi(B) -\Phi(B)\Phi(A)\) for all \(A, B \in \mathcal A\). It is shown that \(\Phi\) has the form \(\Phi(A) = \Psi(A) +\xi (A)\) for every \(A\in\mathcal A\), where \(\Psi\) is an additive isomorphism or the negative of an additive anti-isomorphism and \(\xi:\mathcal A\to \mathbb CI\) is a map satisfying \(\xi(AB - BA) = 0\) for all \(A, B\in \mathcal A\). Here, \(\mathbb C\) denotes the field of complex numbers.
Mathematics Subject Classification ID: 47B49
Mathematics Subject Classification ID: 46L10
zbMATH DE Number: 5285755
zbMATH Keywords: preserver
zbMATH Keywords: Lie product
zbMATH Keywords: von Neumann algebra
reviewed by: Jianlian Cui
MaRDI profile type: Publication
full work available at URL: j.laa.2008.01.031
OpenAlex ID: W2075871686
Wikidata QID: Q115344890
cites work: Commuting Traces of Biadditive Mappings, Commutativity-Preserving Mappings and Lie Mappings
cites work: Jordan mappings of semiprime rings
cites work: Commutativity Preserving Mappings of Von Neumann Algebras
cites work: Commutativity preserving linear maps on central simple algebras.
cites work: Linear maps preserving commutativity
cites work: Q3927697
cites work: Lie Isomorphisms of Simple Rings
cites work: Commutativity Preserving Maps of Factors
cites work: Lie homomorphisms of operator algebras
cites work: Nonlinear commutativity preserving maps on self-adjoint operators
cites work: Preserving commutativity
cites work: Lie Isomorphisms of Factors
cites work: Q5478096
DOI: 10.1016/J.LAA.2008.01.031
description: scientific article; zbMATH DE number 5285755