Adjoining inverses to commutative Banach algebras

Abstract

Let A A be a commutative unital Banach algebra. Suppose G ⊂ A G \subset A is such that | | a | | ⩽ | | g a | | ||a|| \leqslant ||ga|| for all g ∈ G , a ∈ A g \in G,a \in A . Two questions are considered in the paper. Does there exist a superalgebra B B of A A in which every g ∈ G g \in G is invertible? Can one always have also | | g − 1 | | ⩽ 1 ||{g^{ - 1}}|| \leqslant 1 if g ∈ G g \in G ? Arens proved that if G = { g } G = \{ g\} then there is an algebra containing g − 1 {g^{ - 1}} , with | | g − 1 | | ⩽ 1 ||{g^{ - 1}}|| \leqslant 1 . In the paper it is shown that if G G is countable B B exists, but if G G is uncountable, this is not necessarily so. The answer to the second question is negative even if G G consists of only two elements.