Introverted subspaces of the duals of measure algebras

Abstract

Let $\mathcal{G}$ be a locally compact group. In continuation of our studies on the first and second duals of measure algebras by the use of the theory of generalised functions, here we study the C${}^{\ast }$-subalgebra $G{L}_0(\mathcal{G})$ of $GL(\mathcal{G})$ as an introverted subspace of $M(\mathcal{G}{)}^{\ast }$. In the case where $\mathcal{G}$ is non-compact we show that any topological left invariant mean on $GL(\mathcal{G})$ lies in $G{L}_0(\mathcal{G}{)}^{\perp }$. We then endow $G{L}_0(\mathcal{G}{)}^{\ast }$ with an Arens-type product which contains $M(\mathcal{G})$ as a closed subalgebra and $M_a(\mathcal{G})$ as a closed ideal which is a solid set with respect to absolute continuity in $G{L}_0(\mathcal{G}{)}^{\ast }$. Among other things, we prove that $\mathcal{G}$ is compact if and only if $G{L}_0(\mathcal{G}{)}^{\ast }$ has a non-zero left (weakly) completely continuous element.