Mapping and continuity properties of the boundary spectrum in Banach algebras (Q610619)

Let \(A\) be a unital complex Banach algebra and let \(\partial S\) be the boundary of the set of singular elements in \(A\). The boundary spectrum of \(a\in A\) is defined as \(S_\partial(a)=\{\lambda\in \mathbb{C}:\lambda-a\in \partial S\}\); the concept was introduced by the author in earlier work [cf.\ Bull.\ Aust.\ Math.\ Soc.\ 74, No.\,2, 239--246 (2006; Zbl 1113.46044)]. In the paper under review, mapping properties of the boundary spectrum are studied. For instance, it is proved that \(S_\partial\bigl( f(a)\bigr)=f\bigl(S_\partial(a)\bigr)\) if \(f\) is a complex valued function which is analytic and one-to-one on a neighbourhood of \(\sigma(a)\). The last part of the paper is devoted to the continuity properties of the boundary spectrum. It is shown that, for every open set containing \(S_\partial(a)\), there exists \(\delta>0\) such that \(S_\partial(x)\subseteq U\) whenever \(\| a-x\|<\delta\).