## Illinois Journal of Mathematics

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

Wojciech Jaworski

#### 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.

#### Article information

Source
Illinois J. Math., Volume 45, Number 1 (2001), 195-212.

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258138263

Digital Object Identifier
doi:10.1215/ijm/1258138263

Mathematical Reviews number (MathSciNet)
MR1849994

Zentralblatt MATH identifier
0985.43002

#### Citation

Jaworski, Wojciech. Probability measures on almost connected amenable locally compact groups and some related ideals in group algebras. Illinois J. Math. 45 (2001), no. 1, 195--212. doi:10.1215/ijm/1258138263. https://projecteuclid.org/euclid.ijm/1258138263