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Sur le socle dans les algèbres de Jordan-Banach. (On the socle in Jordan-Banach algebras). (French) Zbl 0703.46032

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

MSC:

46H70 Nonassociative topological algebras
17C65 Jordan structures on Banach spaces and algebras
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