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

Given a locally compact group G, let J(G){\cal J}(G) denote the set of closed left ideals in L ¹(G), of the form J _μ = L¹(G) * (δ _e − μ)]⁻, where μ is a probability measure on G. Let J_d(G)={\cal J}_d(G)= {J_m;m is discrete}\{J_{\mu};\mu\ {\rm is discrete}\} , J_a(G)={J_m;m is absolutely continuous}{\cal J}_a(G)=\{J_{\mu};\mu\ {\rm is absolutely continuous}\} . When G is a second countable SIN] group, we prove that J(G)=J_d(G){\cal J}(G)={\cal J}_d(G) and that J_a(G){\cal J}_a(G) , being a proper subset of J(G){\cal J}(G) when G is nondiscrete, contains every maximal element of J(G){\cal J}(G) . Some results concerning the ideals J _μ in general locally compact second countable groups are also obtained.