Hostname: page-component-745bb68f8f-f46jp Total loading time: 0 Render date: 2025-01-26T21:51:58.419Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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. Lecture Notes in Math., Vol. 735 (Springer-Verlag, 1979).CrossRefGoogle Scholar
[2]Aupetit, B.. Inessential elements in Banach algebras. Bull. London Math. Soc. 18 (1986), 493–497.CrossRefGoogle Scholar
[3]Aupetit, B.. A primer on spectral theory. Universitext (Springer-Verlag, 1991).CrossRefGoogle Scholar
[4]Aupetit, B.. Spectral characterization of the radical in Banach and Jordan-Banach algebras. Math. Proc. Camb. Phil. Soc. 114 (1993), 31–35.Google Scholar
[5]Aupetit, B.. Analytic multifunctions and their applications. In Proceedings of the NATO Advanced Study Institute ‘Complex Potential Theory’ (Kluwer Acad. Pub., to appear).Google Scholar
[6]Aupetit, B.. A geometric characterization of algebraic varieties, Proc. Amer. Math. Soc. to appear.Google Scholar
[7]Aupetit, B. and Baribeau, L.. Sur le socle dans les algèbres de Jordan-Banach. Canad. Math. J. 41 (1989), 1090–1100.CrossRefGoogle Scholar
[8]Aupetit, B. and Mouton, H. du T.. Spectrum preserving linear mappings in Banach algebras. Studia Math. 109 (1) (1994), 91–100.Google Scholar
[9]Aupetit, B. and Zraïbi, A.. Propriétés analytiques du spectre dans les algèbres de Jordan-Banach. Manuscripta Math. 38 (1982), 100–105.CrossRefGoogle Scholar
[10]Benslimane, M., Jaa, O. and Kaïdi, A.. The socle and the largest spectrum finite ideal. Quart. J. Math. Oxford 42 (1991), 1–7.Google Scholar
[11]Benslimane, M. and Palacios, A. Rodríguez. Caractérisation spectrale des algèbres de Jordan-Banach non-commutatives complexes modulaires annihilatrices. J. Algebra 140 (1991), 344–354.Google Scholar
[12]López, A. Fernández. Modular annihilator Jordan algebras. Comm. Algebra 13 (1985), 2597–2613.CrossRefGoogle Scholar
[13]López, A. Fernández. Noncommutative Jordan Riesz algebras. Quart. J. Math. Oxford 39 (1988), 67–80.CrossRefGoogle Scholar
[14]López, A. Fern´ndez and Palacios, A. Rodríguez. On the socle of a noncommutative Jordan algebra. Manuscripta Math. 56 (1986), 269–278.CrossRefGoogle Scholar
[15]Loos, O.. Properly algebraic and spectrum-finite ideals in Jordan systems. Math. Proc. Camb. Philos. Soc. 114 (1993), 149–161.CrossRefGoogle Scholar
[16]Moreno, J. Martínez. Sobre álgebras de Jordán normadas completas (Tesis doctoral, Universidad de Granada, 1977).Google Scholar
[17]Moreno, J. Martínez. Holomorphic functional calculus in Jordan-Banach algebras. Ann. Sci. Univ. Biaise Pascal, Clermont II, Sér. Math., Fase. 27 (1991), 125–134.Google Scholar
[18]McCrimmon, K.. A characterization of the radical of a Jordan algebra. J. Algebra 18 (1971), 103–111.CrossRefGoogle Scholar
[19]Osborn, J. M. and Racine, M. L.. Jordan rings with nonzero socle. Trans. Amer. Math. Soc. 215 (1979), 375–387.Google Scholar
[20]Palacios, A. Rodríguez. Jordan structures in analysis. In Proceedings of the 1992 Oberwolfach Conference on Jordan Algebras, to appear.Google Scholar