Non-regularity for Banach function algebras

J. F. Feinstein; D. W. B. Somerset

paper

Non-regularity for Banach function algebras

arXiv:math/9811063

Abstract

Let $A$ be a unital Banach function algebra with character space $Φ_{A}$ . For $x \in Φ_{A}$ , let $M_{x}$ and $J_{x}$ be the ideals of functions vanishing at $x$ , and in a neighbourhood of $x$ , respectively. It is shown that the hull of $J_{x}$ is connected, and that if $x$ does not belong to the Shilov boundary of $A$ then the set ${y \in Φ_{A} : M_{x} \supseteq J_{y}}$ has an infinite, connected subset. Various related results are given.

17 pages plain TeX: expanded introduction and some extra examples