Trace and spectrum preserving linear mappings in Jordan-Banach algebras
\textit{B. Aupetit} and \textit{H. du T. Mouton} [Stud. Math. 121, No. 2, 115-136 (1996; Zbl 0872.46028)] defined the trace and the determinant for elements of the socle of a semiprimitive Banach algebra in a purely spectral and analytic way. Then they used the analytic properties of the spectrum to prove that the trace and the determinant are entire functions, and to deduce their basic properties. Later, these authors together with A. Maouche extended these results to Jordan Banach algebras. In the first part of the paper under review, further identities concerning the trace defined on the socle of a semiprimitive Jordan Banach algebra \(J\) are obtained, and used to prove that \(\text{Tr} (ay)\), for \(a\) in \(J\) and \(y\) in \(\text{Soc}(J)\), defines a bilinear form which coincides with the Faulkner form when restricted to the socle. Moreover, the annihilator of the socle coincides with the orthogonal of the socle. The reader is referred to a recent paper of \textit{O. Loos} [Commun. Algebra 25, No. 9, 3011-3042 (1997)] for a purely algebraic approach to this question. Using the results of the first section, the author proves that, under certain conditions, spectrum preserving linear mappings between Jordan Banach algebras are actually homomorphisms.
scientific article; zbMATH DE number 1154519
scientific article