Probability measures on almost connected amenable locally compact groups and some related ideals in group algebras

Abstract

Given a locally compact group $G$ let $\mathcal{J}_a(G)$ denote the set of all closed left ideals $J$ in $L^1(G)$ which have the form $J=[L^1(G)*(\delta_e -\mu)]\overline{\vphantom{t}\ }$ where $\mu$ is an absolutely continuous probability measure on $G$. We explore the order structure of $\mathcal{J}_a(G)$ when $\mathcal{J}_a(G)$ is ordered by inclusion. When $G$ is connected and amenable we prove that every nonempty family $\mathcal{F}\subseteq \mathcal{J}_a(G)$ admits both a minimal and a maximal element; in particular, every ideal in $\mathcal{J}_a(G)$ contains an ideal that is minimal in $\mathcal{J}_a(G)$. Furthermore, we obtain that every chain in $\mathcal{J}_a(G)$ is necessarily finite. A natural generalization of these results to almost connected amenable groups is discussed. Our proofs use results from the theory of boundaries of random walks.