On bilinear maps determined by rank one idempotents
type: Item
instance of: scholarly article
title: On bilinear maps determined by rank one idempotents
zbMATH Open document ID: 1185.15024
author: J. Extremera
author: Jerónimo Alaminos Prats
author: Matej Brešar
author: Armando R. Villena
published in: Linear Algebra and its Applications
publication date: 2009-12-18T00:00:00Z
review text: Using \(C^\ast\)-algebra techniques, the authors proved in [Stud. Math. 193, No. 2, 131--159 (2009; Zbl 1168.47029)] that for the algebra of square complex matrices and for a complex linear space \(X\), any bilinear map \(\phi:A\times A\rightarrow X\) is of the form \(\phi(a,b)=\phi_1(ab)+\phi_2(ba)\), where \(\phi_1,\phi_2:A\rightarrow X\) are linear maps, in case \(\phi(a,b)=0\) whenever \(ab=ba=0\). In this paper, based on elementary calculations, this result is extended to the algebra of all square matrices of a fixed order, over a field of characteristic different from \(2\), under the assumption \(\phi(e,f)=0\) whenever \(e\) and \(f\) are rank one idempotents that are orthogonal. Applications of this result to some linear preserver problems are presented.
Mathematics Subject Classification ID: 15A86
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15A03
Mathematics Subject Classification ID: 15A27
Mathematics Subject Classification ID: 15A30
zbMATH DE Number: 5650007
zbMATH Keywords: matrix algebra
zbMATH Keywords: zero product
zbMATH Keywords: rank one idempotent
zbMATH Keywords: linear map
zbMATH Keywords: bilinear map
zbMATH Keywords: linear preserver problem
reviewed by: C. M.da Fonseca
MaRDI profile type: Publication
full work available at URL: j.laa.2009.09.017
OpenAlex ID: W1981343147
cites work: Characterizing homomorphisms and derivations on C*-algebras
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DOI: 10.1016/J.LAA.2009.09.017
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description: scientific article; zbMATH DE number 5650007