Hereditary properties of spectral isometries
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.
scientific article; zbMATH DE number 2119259