Maps on spaces of linear transformations over semisimple algebras
type: Item
instance of: scholarly article
title: Maps on spaces of linear transformations over semisimple algebras
zbMATH Open document ID: 0658.16015
DOI: 10.1016/0021-8693(88)90265-7
author: Warren J. Wong
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: It is well-known that for any field K every bijective linear map f on \(M_ n(K)\) which preserves the determinant function is an equivalence transformation \((f(A)=PAQ\) for some invertible matrices P and Q) or the composite of an equivalence transformation with the transpose transformation \((f(A)=A^ T)\). A theorem of \textit{J. Dieudonné} [Arch. Math. 1, 282-287 (1949; Zbl 0032.10601)] shows that, in fact, this is true for the linear map which preserves the set of singular matrices, and \textit{M. Marcus} and \textit{B. Moyls} [Pac. J. Math. 9, 1215-1221 (1959; Zbl 0089.089)] showed that the same holds for the bijective linear maps which preserve the set of matrices of rank 1. In this paper the author extends the problem to maps on spaces of linear transformations over semisimple algebras, preserving transformations of local length 1, a notion defined by the author as an extension of the property of having rank 1. He also generalizes Dieudonné's theorem by determining the maps preserving the set of ``locally singular'' transformations.
reviewed by: Xu Yonghua
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 16Kxx
zbMATH DE Number: 4075317
zbMATH Keywords: determinant function
zbMATH Keywords: equivalence transformation
zbMATH Keywords: transpose transformation
zbMATH Keywords: linear maps
zbMATH Keywords: linear transformations over semisimple algebras
zbMATH Keywords: local length
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90265-7
OpenAlex ID: W2004218540
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description: scientific article; zbMATH DE number 4075317