Illinois Journal of Mathematics

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

Wojciech Jaworski

Full-text: Open access

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

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258138263

Digital Object Identifier
doi:10.1215/ijm/1258138263

Mathematical Reviews number (MathSciNet)
MR1849994

Zentralblatt MATH identifier
0985.43002

Subjects
Primary: 43A05: Measures on groups and semigroups, etc.
Secondary: 43A07: Means on groups, semigroups, etc.; amenable groups 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization

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


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