Rank-preserving multiplicative maps on \(B(X)\)
type: Item
instance of: scholarly article
title: Rank-preserving multiplicative maps on \(B(X)\)
zbMATH Open document ID: 1038.47028
DOI: 10.1016/S0024-3795(01)00434-7
author: Guimei An
author: Jinchuan Hou
published in: Linear Algebra and its Applications
publication date: 2002-05-15T00:00:00Z
full work available at URL: abstract.html
review text: Let \(X\) (resp., \(H\)) be a Banach space (resp., Hilbert space). By \({\mathcal B}(X)\) (resp., \({\mathcal B}(H))\) denote the algebra of all bounded linear operators on \(X\) (resp., \(H\)). The authors describe the structure of multiplicative maps preserving rank on \({\mathcal B}(X)\). Furthermore, they prove that every multiplicative local approximate automorphism of \({\mathcal B}(X)\) the range of which contains all rank-1 idempotents is in fact an automorphism. They also characterize multiplicative maps preserving co-rank on \({\mathcal B}(H)\) and prove that a multiplicative map \(\Phi :{\mathcal B}(H)\rightarrow {\mathcal B}(K)\) the range of which contains all rank-1 projections preserves the property \(A^*B=0\Leftrightarrow \Phi (A)^*\Phi (B)=0\) for any \(A\), \(B\in {\mathcal B}(H)\) if and only if there exists a unitary or conjugate linear unitary operator \(U\in {\mathcal B}(H, K)\) such that \(\Phi (A)=UAU^*\) for all \(A\in {\mathcal B}(H)\).
Mathematics Subject Classification ID: 47B49
Mathematics Subject Classification ID: 46H99
zbMATH DE Number: 1741560
zbMATH Keywords: rank
zbMATH Keywords: co-rank
zbMATH Keywords: multiplicative map
zbMATH Keywords: isomorphism
reviewed by: Jianlian Cui
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1741560