Weak*-Simplicity of Convolution Algebras on Discrete Groups

Abstract

We prove that, given a discrete group $G$, and $1\le p<\infty$, the algebra of $p$-convolution operators $C{V}_p(G)$ is weak*-simple, in the sense of having no non-trivial weak*-closed ideals, if and only if $G$ is an ICC group. This generalises the basic fact that $vN(G)$ is a factor if and only if $G$ is ICC. When $p=1$, $C{V}_p(G)={\ell }^1(G)$. In this case we give a more detailed analysis of the weak*-closed ideals, showing that they can be described in terms of the weak*-closed ideals of ${\ell }^1(FC(G))$; when $FC(G)$ is finite, this leads to a classification of the weak*-closed ideals of ${\ell }^1(G)$.

18 pages. To appear in Studia Mathematica