Jordan maps on triangular algebras.
type: Item
instance of: scholarly article
title: Jordan maps on triangular algebras.
zbMATH Open document ID: 1127.16029
published in: Linear Algebra and its Applications
publication date: 2007-09-13T00:00:00Z
review text: For \(x,y\in R\), an associative ring, let \(x\circ y=xy+yx\). Given rings \(R\) and \(S\), maps \(f\colon R\to S\) and \(g\colon S\to R\) form a Jordan pair if for all \(x\in R\) and \(y\in S\), \(f(x\circ g(y))=f(x)\circ y\) and \(g(y\circ f(x))=g(y)\circ x\). The author shows that in certain cases each of \(f\) and \(g\) forming a Jordan pair must be additive maps. Necessary conditions for this conclusion are that both \(f\) and \(g\) are surjective, and that one of \(R\) or \(S\) is a faithful triangular algebra. Such an algebra is a \(2\times 2\) triangular matrix ring with upper and lower diagonal entries from algebras \(A\) and \(B\), respectively, over a commutative ring \(R\), so that \(a\circ A=0\) for \(a\in A\) forces \(a=0\) and \(b\circ B=0\) for \(b\in B\) forces \(b=0\), and with the upper right entries coming from a faithful \((A,B)\) module \(X\) so that neither \(A\) nor \(B\) annihilates any nonzero \(x\in X\).
reviewed by: Charles Lanski
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 46L40
Mathematics Subject Classification ID: 15A78
zbMATH DE Number: 5190769
zbMATH Keywords: Jordan maps
zbMATH Keywords: triangular algebras
zbMATH Keywords: additivity
zbMATH Keywords: Jordan pairs
zbMATH Keywords: additive maps
zbMATH Keywords: triangular matrix rings
author: Peisheng Ji
MaRDI profile type: Publication
full work available at URL: j.laa.2007.04.009
OpenAlex ID: W2032135072
cites work: Commuting Maps of Triangular Algebras
cites work: Lie Derivations of Triangular Algebras
cites work: Additivity of Jordan | ast-Maps on AW ∗ -Algebras
cites work: Additivity of Jordan *-maps between operator algebras
cites work: Jordan maps of nest algebras
cites work: Additivity of Jordan maps on standard operator algebras
cites work: Jordan Maps on Associative Algebras
cites work: Multiplicative mappings of operator algebras
cites work: When are Multiplicative Mappings Additive?
cites work: Jordan derivations of triangular algebras
DOI: 10.1016/J.LAA.2007.04.009
description: scientific article; zbMATH DE number 5190769