Probability measures on [SIN] groups and some related ideals in group algebras

Abstract

Given a locally compact group G, let \({\cal J}(G)\) denote the set of closed left ideals in L1(G), of the form Jμ = [L1(G) * (δe − μ)]−, where μ is a probability measure on G. Let \({\cal J}_d(G)=\)\(\{J_{\mu};\mu\ {\rm is discrete}\}\), \({\cal J}_a(G)=\{J_{\mu};\mu\ {\rm is absolutely continuous}\}\). When G is a second countable [SIN] group, we prove that \({\cal J}(G)={\cal J}_d(G)\) and that \({\cal J}_a(G)\), being a proper subset of \({\cal J}(G)\) when G is nondiscrete, contains every maximal element of \({\cal J}(G)\). Some results concerning the ideals Jμ in general locally compact second countable groups are also obtained.