Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T01:28:37.677Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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