Maps preserving semi-Fredholm operators on Hilbert \(C^*\)-modules
type: Item
instance of: scholarly article
title: Maps preserving semi-Fredholm operators on Hilbert \(C^*\)-modules
zbMATH Open document ID: 1173.46037
author: T. Aghasizadeh
author: Shirin Hejazian
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-05-19T00:00:00Z
review text: Let \(M\) denote a Hilbert \(C^*\)-module over a \(C^*\)-algebra \(A\). Let \(L(M)\) and \(K(M)\) be the \(C^*\)-algebras of all adjointable linear operators and compact operators on \(M\), respectively. The authors prove that if \(\varphi: L(M) \to L(M)\) is a linear map preserving semi-\(A\)-Fredholm operators in both directions, \(\varphi(I) = I\), and \(\varphi\) is surjective up to compact operators, then \(\varphi(K(M))\subseteq K(M)\). Furthermore, they show that the induced map \(\widetilde{\varphi}: L(M)/K(M) \to L(M)/K(M)\) is either an automorphism or an anti-automorphism, if one of the following conditions holds: (i) \(M\) is a \(K(H)\)-Hilbert module; (ii) \(M\) is a Hilbert \(C^*\)-module such that \(L(M)\) has the FS-property (i.e., the set of self-adjoint elements of \(L(M)\) being of finite spectrum is norm dense in the set of self-adjoint elements) and \(K(M)\) has a prime corona algebra; (iii) \(A\) is a \(W^*\)-algebra and \(M\) is a self-dual Hilbert module over \(A\) such that \(K(M)\) has a prime corona algebra. These results generalize Theorem 1.2 of \textit{M.\,Mbekhta} and \textit{P.\,Šemrl} [Linear Multilinear Algebra 57, No.\,1, 55--64 (2009; Zbl 1201.47015)].
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 46L08
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 47A53
zbMATH DE Number: 5555219
zbMATH Keywords: Hilbert \(C^*\)-module
zbMATH Keywords: semi-\(A\)-Fredholm operator
zbMATH Keywords: Calkin algebra
zbMATH Keywords: preserving map
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.01.032
OpenAlex ID: W2149761179
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DOI: 10.1016/J.JMAA.2009.01.032
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description: scientific article; zbMATH DE number 5555219