Inverse producing extension of a Banach algebra which eliminates the residual spectrum of one element

Abstract

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