Aupetit, Bernard; Baribeau, Line Sur le socle dans les algèbres de Jordan-Banach. (On the socle in Jordan-Banach algebras). (French) Zbl 0703.46032 Can. J. Math. 41, No. 6, 1090-1100 (1989). The socle of a Banach algebra is an analogue of the set of operators of finite rank in operator algebras on a Banach space. In previous papers of B. A. Barnes [Trans. Am. Math. Soc. 133, 511-517 (1968; Zbl 0159.185)] and B. Aupetit [Proprietés spectrales des algèbres de Banach, Lect. Notes Math. 735 (1979; Zbl 0409.46054)] some sufficient conditions were given for the socle to be non zero. The main purpose of the present paper is to generalize these results to Jordan Banach algebras. A Jordan Banach algebra J is said to be non degenerate if \(U_ xy=0\) for all \(y\in J\) implies \(x=0\). The socle of a non degenerate Jordan algebra is the sum of all minimal quadratic ideals of J. The main result is the following Theorem 11. Let J be a Jordan Banach algebra with identity and without radical. Suppose that every element of J has at most countable spectrum. Then the socle of J is non zero. Reviewer: Sh.A.Ayupov Cited in 4 Documents MSC: 46H70 Nonassociative topological algebras 17C65 Jordan structures on Banach spaces and algebras Keywords:spectrum; socle of a Banach algebra; set of operators of finite rank in operator algebras on a Banach space; Jordan Banach algebras; socle of a non degenerate Jordan algebra; sum of all minimal quadratic ideals Citations:Zbl 0159.185; Zbl 0409.46054 PDFBibTeX XMLCite \textit{B. Aupetit} and \textit{L. Baribeau}, Can. J. Math. 41, No. 6, 1090--1100 (1989; Zbl 0703.46032) Full Text: DOI