On the invariant mean on topological semigroups and on topological groups

Edmond E. Granirer

Vol. 15 (1965), No. 1, 107–140

DOI: 10.2140/pjm.1965.15.107

Abstract

Let $S$ be a topological semigroup and $C\left(S\right)$ be the space of bounded continous functions on $S$. The space of translation invariant, bounded, linear functionals on $C\left(S\right)$ and its connection with the structure of $S$, are investigated in this paper. For topological groups $G$, not necessarily locally compact, the space of bounded, linear, translation invariant functionals, on the space $UC\left(G\right)$ of bounded uniformly continuous functions, is also investigated and its connection with the structure of $G$ pointed out. The obtained results are applied to the study of the radical of the convolution algebra $UC{\left(G\right)}^{\ast }$ (for locally compact groups, or for subgroups of locally convex linear topological spaces) and some results which seem to be unknown even when $G$ is taken to be the real line are obtained.

Mathematical Subject Classification

Primary: 22.10

Milestones

Received: 2 December 1963

Published: 1 March 1965