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 generalized functions, here we study the C$^*$-subalgebra $GL_0(\mathcal{G})$ of $GL(\mathcal{G})$ as an introverted subspace of $M(\mathcal{G} )^*$. In the case where $\mathcal{G} $ is non-compact, we show that any topological left invariant mean on $GL(\mathcal{G} )$ lies in $GL_0(\mathcal{G} )^\perp $. We then endow $GL_0(\mathcal{G} )^*$ 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 $GL_0(\mathcal{G} )^*$. Among other things, we prove that $\mathcal{G} $ is compact if and only if $GL_0(\mathcal{G} )^*$ has a non-zero left (weakly) completely continuous element.