Weak$^*$-simplicity of convolution algebras on discrete groups
Volume 278 / 2024
Studia Mathematica 278 (2024), 267-289
MSC: Primary 43A15; Secondary 43A20, 47L10, 46H10
DOI: 10.4064/sm231214-9-9
Published online: 6 November 2024
Abstract
We prove that, given a discrete group $G$, and $1 \leq p \lt \infty $, the algebra of $p$-convolution operators $\mathrm {CV}_p(G)$ 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 $\mathrm {vN}(G)$ is a factor if and only if $G$ is ICC. When $p=1$, $\mathrm {CV}_p(G) = \ell ^1(G)$. 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 $\ell ^1(\mathrm {FC}(G))$; when $\mathrm {FC}(G)$ is finite, this leads to a classification of the weak$^*$-closed ideals of $\ell ^1(G)$.
