Inverse Producing Extension of a Banach Algebra which Eliminates the Residual Spectrum of One Element

Abstract

If $A$ is a commutative unital Banach algebra and $G \subset A$ is a collection of nontopological zero divisors, the question arises whether we can find an extension $A\prime$ of $A$ in which every element of $G$ has an inverse. Shilov [1] proved that this was the case if $G$ consisted of a single element, and Arens [2] conjectures that it might be true for any set $G$. In [3], Bollobás proved that this is not the case, and gave an example of an uncountable set $G$ for which no extension $A\prime$ can contain inverses for more than countably many elements of $G$. Bollobás proved that it was possible to find inverses for any countable $G$, and gave best possible bounds for the norms of the inverses in [4]. In this paper, it is proved that inverses can always be found if the elements of $G$ differ only by multiples of the unit; that is, we can eliminate the residual spectrum of one element of $A$. This answers the question posed by Bollobás in [5].