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Spectral characterization of the socle in Jordan–Banach algebras

Published online by Cambridge University Press:  24 October 2008

Bernard Aupetit
Bernard Aupetit
Affiliation:
Département de mathématiques et de statistique, Université Laval, Québec, Canada, G1K 7P4
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Extract

If A is a complex Banach algebra the socle, denoted by Soc A, is by definition the sum of all minimal left (resp. right) ideals of A. Equivalently the socle is the sum of all left ideals (resp. right ideals) of the form Ap (resp. pA) where p is a minimal idempotent, that is p2 = p and pAp = ℂp. If A is finite-dimensional then A coincides with its socle. If A = B(X), the algebra of bounded operators on a Banach space X, the socle of A consists of finite-rank operators. For more details about the socle see [1], pp. 78–87 and [3], pp. 110–113.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 117 , Issue 3 , May 1995 , pp. 479 - 489
Copyright
Copyright © Cambridge Philosophical Society 1995

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References

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