An Analytic Characterization of Groups with no Finite Conjugacy Classes

Abstract

Let $A$ be a unital Banach algebra and $\mathcal {G}$ the group of isometries in $A$. The norm in $A$ is uniquely maximal if $\mathcal {G}$ is not contained in any larger bounded group in $A$ and there is no equivalent norm on $A$ with the same group of isometries. We use a group theory result of B. H. Neumann to prove that the discrete measure algebra ${l^1}(G)$ is uniquely maximal if and only if $G$ has no finite conjugacy classes.