Weak*-Simplicity of Convolution Algebras on Discrete Groups

TL;DR

This paper characterizes when the algebra of p-convolution operators on a discrete group is weak*-simple, linking it to the group being ICC, and provides a detailed analysis of ideals in the case p=1.

Contribution

It establishes a characterization of weak*-simplicity of convolution algebras on discrete groups based on ICC property, extending known results for von Neumann algebras.

Findings

CV_p(G) is weak*-simple iff G is ICC.

Weak*-closed ideals of (G) relate to those of (FC(G)).

Classification of ideals when FC(G) is finite.

Abstract

We prove that, given a discrete group $G$, and $1 \leq p < \infty$, the algebra of $p$-convolution operators $CV_p(G)$ 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 $p=1$, $CV_p(G) = \ell^1(G)$. 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 $\ell^1(G)$.