Abstract
In the first section we define the trace on the socle of a Jordan-Banach algebra in a purely spectral way and we prove that it satisfies several identities. In particular this trace defines the Faulkner bilinear form. In the second section, using analytic tools and the properties of the trace, we prove that a spectrum preserving linear mapping fromJ ontoJ 1, whereJ andJ 1 are semisimple Jordan-Banach algebras, is not far from being a Jordan isomorphism. It is in particular a Jordan isomorphism ifJ 1 is primitive with non-zero socle.
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References
Aupetit B (1986) Inessential elements in Banach algebras. Bull London Math Soc18: 493–497
Aupetit B (1994) Spectral characterization of the radical in Banach and Jordan-Banach algebras. AMath Proc Camb Phil Soc114: 31–35
Aupetit B (1994) Recent trends in the field of Jordan-Banach algebras. In:Zemánek J (ed) Functional Analysis and Operator Theory, vol 30. Polish Academy of Sciences: Banach Center Publ
Aupetit B (1994) Analytic Multifunctions and Their Applications. In:Gauthier PM (ed) Complex Potential Theory. Kluwer
Aupetit B (1995) Spectral characterization of the socle in Jordan-Banach algebras. Math Proc Camb Phil Soc117: 479–489
Aupetit B, Baribeau L (1989) Sur le socle dans les algébres de Jordan-Banach. Can J Math41: 1090–1100
Aupetit B, Mouton H du T. (1994) Spectrum preserving linear mappings in Banach algebras. Studia Math109: 91–100
Aupetit B, Mouton H du T. (1996) Trace and determinant in Banach algebras. Studia Math121: 195–200
Aupetit B, Maouche A, Mouton H du T. (preprint) Trace and determinant in Jordan-Banach algebras
Fernández López A (1992) On annihilators in Jordan algebras. Publ Matemàtiques36: 569–589
Fernández López A, Rodríguez Palacios A (1986) On the socle of a non-commutatives Jordan algebra. Manuscripta Math56: 269–278
Fernández López A, Rodríguez Palacios A (1986) Primitive non-commutative Jordan algebras with non-zero socle. Proc Amer Math Soc96: 199–206
Maouche A (1994) Théorie spectrale dans les systèmes de Jordan-Banach. Thesis, Univ Laval, Québec
McCrimmon K (1971) A characterization of the radical of a Jordan algebra. J Algebra18: 101–111
McCrimmon K (1985) Reduced elements in Jordan triple systems. J Algebra97: 540–564
Wilkins TJD (1997) Inessential ideals in Jordan-Banach algebras. Bull London Math Soc136: 73–81
Zel'manov E (1987) Goldie's theorem for Jordan algebras. Siberian Math J28: 44–52
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Aupetit, B. Trace and spectrum preserving linear mappings in Jordan-Banach algebras. Monatshefte für Mathematik 125, 179–187 (1998). https://doi.org/10.1007/BF01317312
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DOI: https://doi.org/10.1007/BF01317312