Trace and spectrum preserving linear mappings in Jordan-Banach algebras
type: Item
instance of: scholarly article
title: Trace and spectrum preserving linear mappings in Jordan-Banach algebras
zbMATH Open document ID: 0919.46037
DOI: 10.1007/BF01317312
author: Bernard Aupetit
published in: Monatshefte für Mathematik
publication date: 1999-06-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178806
review text: \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.
reviewed by: Antonio Fernández López
Mathematics Subject Classification ID: 46H70
Mathematics Subject Classification ID: 17C65
zbMATH DE Number: 1154519
zbMATH Keywords: spectrum preserving mapping
zbMATH Keywords: socle
zbMATH Keywords: trace
zbMATH Keywords: determinant
zbMATH Keywords: socle of a semiprimitive Banach algebra
zbMATH Keywords: Jordan Banach algebras
zbMATH Keywords: semiprimitive Jordan Banach algebra
zbMATH Keywords: homomorphisms
MaRDI profile type: Publication
cites work: Spectral characterization of the radical in Banach and Jordan–Banach algebras
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cites work: Spectrum preserving linear mappings in Banach algebras
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cites work: Trace and determinant in Jordan-Banach algebras
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description: scientific article; zbMATH DE number 1154519