# Introverted subspaces of the duals of measure algebras

## TL;DR

This paper investigates the structure of certain subspaces of measure algebra duals for locally compact groups, revealing conditions under which the group is compact based on properties of these subspaces.

## Contribution

It introduces and analyzes the introverted subspace $GL_0({
mf G})$ of measure algebra duals, establishing new links between group compactness and algebraic properties.

## Key findings

- Any topological left invariant mean on $GL({
mf G})$ lies in $GL_0({
mf G})^ot$ for non-compact groups.
- The dual $GL_0({
mf G})^*$ admits an Arens-type product containing $M({
mf G})$ as a closed subalgebra.
- ${
mf G}$ is compact iff $GL_0({
mf G})^*$ has a non-zero left (weakly) completely continuous element.

## 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.