Skip to main content
×
Home
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 4
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Costara, Constantin 2014. Local spectrum linear preservers at non-fixed vectors. Linear Algebra and its Applications, Vol. 457, p. 154.


    Braatvedt, G. and Brits, R. 2013. Uniqueness and spectral variation in Banach algebras. Quaestiones Mathematicae, Vol. 36, Issue. 2, p. 155.


    Mathieu, Martin 2009. A COLLECTION OF PROBLEMS ON SPECTRALLY BOUNDED OPERATORS. Asian-European Journal of Mathematics, Vol. 02, Issue. 03, p. 487.


    Aupetit, Bernard 1995. Spectral characterization of the socle in Jordan–Banach algebras. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 117, Issue. 03, p. 479.


    ×
  • 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: http://dx.doi.org/10.1017/S0305004100071371
  • Published online: 24 October 2008
Abstract

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.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[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.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×