Hereditary properties of spectral isometries
type: Item
instance of: scholarly article
title: Hereditary properties of spectral isometries
zbMATH Open document ID: 1075.47022
author: Martin Mathieu
author: A. R. Sourour
published in: Archiv der Mathematik
publication date: 2004-11-26T00:00:00Z
review text: Let \(A\) and \(B\) be unital semisimple Banach algebras over the complex field. A linear mapping \(T: A \to B\) is called a spectral isometry if \(T\) preserves the spectral radius of every element, that is, \(r(T(x)) = r(x)\) for all \(x\in A\). \textit{B. Aupetit} proved in [``A primer on spectral theory'' (Universitext, Springer--Verlag, Berlin--Heidelberg--New York) (1991; Zbl 0715.46023)] that every spectral isometry \(T\) is injective. Moreover, \(T\) is bounded whenever it is assumed to be surjective. As a consequence of Pták's characterization of the centre, it can be shown that every surjective spectral isometry maps the centre \(Z(A)\) of \(A\) onto the centre \(Z(B)\) of \(B\). From the above facts and the classical Nagasawa's theorem, \textit{B. Aupetit} deduced in [op. cit.] that \(T| _{Z(A)}\) is an algebra isomorphism from \(Z(A)\) onto \(Z(B)\). In the previous years, a lot of attention has been paid to the problem of what additional hypothesis (on \(T\) or on the algebras \(A\) and \(B\)) could be sufficient to show that every unital surjective spectral isometry is a Jordan isomorphism. In [Stud. Math. 152, No. 2, 187--199 (2002; Zbl 1030.47026)], \textit{M. Mathieu} and \textit{G. Schick} conjectured that this holds whenever \(A\) and \(B\) are C*-algebras. Up to this moment, this is an open problem. In the paper under review, the authors give a positive answer to the above problem in some subclass of von Neumann algebras. In their main result, they prove that if \(T\) is a unital surjective spectral isometry between two von Neumann algebras \(A\) and \(B,\) then \(T\) is a Jordan isomorphism whenever \(A\) or \(B\) is a type I von Neumann algebra.
reviewed by: Antonio M. Peralta
Mathematics Subject Classification ID: 47B48
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L70
Mathematics Subject Classification ID: 47A65
Mathematics Subject Classification ID: 17C65
zbMATH DE Number: 2119259
zbMATH Keywords: spectral isometries
zbMATH Keywords: type I von Neumann algebras
MaRDI profile type: Publication
full work available at URL: s00013-003-0595-5
OpenAlex ID: W2012006723
DOI: 10.1007/S00013-003-0595-5
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description: scientific article; zbMATH DE number 2119259