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  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 114, Issue 1
  • July 1993, pp. 31-35

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

  • Bernard Aupetit (a1)
  • DOI:
  • Published online: 24 October 2008

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.

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[1]B. Aupetit . Propriétés spectrales des algèbres de Banach (Springer-Verlag, 1979).

[2]B. Aupetit . The uniqueness of the complete norm topology in Banach algebras and Banach–Jordan algebras. J. Funci. Anal. 47 (1982), 16.

[3]B. Aupetit . A Primer on Spectral Theory (Springer-Verlag, 1991).

[4]B. Aupetit and J. Zemánek . Local characterizations of the radical in Banach algebras. Bull. London Math. Soc. 15 (1983), 2530.

[5]B. Aupetit and A. Zraibi . Propriétés analytiques du spectre dans les algèbres de Jordan–Banach. Manuscripta Math. 38 (1982), 381386.

[8]S. Perlis . A characterization of the radical of an algebra. Bull. Amer. Math. Soc. 48 (1942), 128132.

[9]T. Ransford and M. Write . Holomorphic self-maps of the spectral unit ball. Bull. London Math. Soc. 23 (1991), 256262.

[10]A. Rodríguez-Palacios . The uniqueness of the complete norm topology in complete normed non-associative algebras. J. Funct. Anal. 60 (1985), 115.

[12]J. Zemánek . A note on the radical of a Banach algebra. Manuscript Math. 20 (1977), 191196.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
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