Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

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

Bernard Aupetita1

a1 Département de Mathématiques et de Statistique, Faculté des Sciences et de Genie, Université Laval, Québec, Canada G1K 7P4

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.

(Received June 04 1992)

(Revised October 21 1992)