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$^*$-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.