Adjoining inverses to commutative Banach algebras

Abstract:

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