Continuity of convolution and SIN groups

TL;DR

This paper investigates the continuity properties of convolution in measure algebras of topological groups, establishing that joint continuity occurs precisely when the group has the SIN property.

Contribution

It characterizes the exact topological group condition (SIN property) under which convolution becomes jointly continuous in the measure algebra.

Findings

Convolution is separately continuous on the measure algebra.

Joint continuity of convolution is equivalent to the group having the SIN property.

Abstract

Let the measure algebra of a topological group be equipped with the topology of uniform convergence on bounded right uniformly equicontinuous sets of functions. Convolution is separately continuous on the measure algebra, and it is jointly continuous if and only if the group has the SIN property.