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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

REFERENCES

[1]Aupetit, B.. Propriétés spectrales des algèbres de Banach (Springer-Verlag, 1979).CrossRefGoogle Scholar
[2]Aupetit, B.. The uniqueness of the complete norm topology in Banach algebras and Banach–Jordan algebras. J. Funci. Anal. 47 (1982), 1–6.CrossRefGoogle Scholar
[3]Aupetit, B.. A Primer on Spectral Theory (Springer-Verlag, 1991).CrossRefGoogle Scholar
[4]Aupetit, B. and Zemánek, J.. Local characterizations of the radical in Banach algebras. Bull. London Math. Soc. 15 (1983), 25–30.CrossRefGoogle Scholar
[5]Aupetit, B. and Zraibi, A.. Propriétés analytiques du spectre dans les algèbres de Jordan–Banach. Manuscripta Math. 38 (1982), 381–386.CrossRefGoogle Scholar
[6]Hanche-Olsen, H. and Størmer, E.. Jordan Operator Algebras (Pitman, 1984).Google Scholar
[7]McCrimmon, K.. The radical of a Jordan algebra. Proc. Nat. Acad. Sci. U.S.A. 59 (1969), 671–678.CrossRefGoogle Scholar
[8]Perlis, S.. A characterization of the radical of an algebra. Bull. Amer. Math. Soc. 48 (1942), 128–132.CrossRefGoogle Scholar
[9]Ransford, T. and Write, M.. Holomorphic self-maps of the spectral unit ball. Bull. London Math. Soc. 23 (1991), 256–262.CrossRefGoogle Scholar
[10]Rodríguez-Palacios, A.. The uniqueness of the complete norm topology in complete normed non-associative algebras. J. Funct. Anal. 60 (1985), 1–15.CrossRefGoogle Scholar
[11]Rodríguez-Palacios, A.. Automatic continuity with application to C* algebras. Math. Proc. Cambridge Philos. Soc. 107 (1990), 345–347.Google Scholar
[12]Zemánek, J.. A note on the radical of a Banach algebra. Manuscript Math. 20 (1977), 191–196.CrossRefGoogle Scholar
[13]Zhevlakov, K. A., Slin'ko, A. M., Shestakov, I. P. and Shirshov, A. I.. Rings That Are Nearly Associative (Academic Press, 1982).Google Scholar