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Spectral characterization of the radical in Banach and Jordan–Banach algebras

Published online by Cambridge University Press:  24 October 2008

Bernard Aupetit
Bernard Aupetit
Affiliation:
Département de Mathématiques et de Statistique, Faculté des Sciences et de Genie, Université Laval, Québec, CanadaG1K 7P4
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Extract

If a is a n × n matrix such that a + m is invertible for every invertible a + m matrix m, then a = 0, by a classical result of Perlis [8]. Unfortunately the same result is not true in general for semi-simple rings as shown by T. Laffey. In the general situation of Banach algebras, Zemánek[12] has proved that a is in the Jacobson radical of A if and only if ρ(a+x) = ρ(x), for every x in A, where ρ denotes the spectral radius. More sophisticated characterizations of the radical were given in [4] and [3], theorem 5·3·1. The arguments used in all these situations depend heavily on representation theory.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 1 , July 1993 , pp. 31 - 35
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

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