Mapping and continuity properties of the boundary spectrum in Banach algebras

Abstract

We present further properties of the boundary spectrum $S_{\partial }(a) = \{\lambda: \lambda-a \in\partial S\}$ of $a$, where $\partial S$ denotes the topological boundary of the set $S$ of all noninvertible elements of a Banach algebra $A$, and where $a$ is an element of $A$. In particular, we investigate the conditions under which it is true that $S_{\partial}(f(a))=f(S_{\partial}(a))$, where $f$ is a complex valued function which is analytic on a neighbourhood of the spectrum of $a$. We also consider continuity properties of the boundary spectrum.