Continuity of spectrum and spectral radius in Banach algebras

Abstract

This survey deals with necessary and/or sufficient conditions for continuity of the spectrum and spectral radius functions at a point of a Banach algebra. Introduction. Let L be a complex Banach algebra. If L has no identity, let L denote the Banach algebra obtained by canonical adjunction of an identity to L, whereas we set L = L if L has an identity. Furthermore, let KC denote the set of compact nonempty subsets of the complex plane C, endowed with the Hausdorff metric ∆. We deal with continuity of the spectrum function σ : L → KC and of the spectral radius function r : L → R (where σ(a) and r(a) denote respectively the spectrum and the spectral radius of a in L for any a ∈ L). The following inequality is not difficult to verify: (1) |r(a)− r(b)| ≤ ∆(σ(a) , σ(b)) for any a, b ∈ L. Hence continuity of σ implies continuity of r. If Ω is a subset of a topological space Ξ, we denote by Ω and Ω ◦ the closure and interior of Ω in Ξ, respectively. If L is commutative, by the Gelfand representation theorem (see [R], 3.1.6, 3.1.11 and 3.1.20) there exist a locally compact Hausdorff space X and a continuous homomorphism Γ , from L into the Banach algebra C0(X) of complex-valued continuous functions on X which vanish at infinity, such that σ(a) = â(X) (where 1991 Mathematics Subject Classification: Primary 46H99. The paper is in final form and no version of it will be published elsewhere.