The Criterion of Completely Reducibility for Continuous Representations of Group Algebras

TL;DR

This paper establishes that all nonsingular continuous representations of the group algebra $L^{1}(G)$ are completely reducible precisely when the underlying group $G$ is compact, linking algebraic properties to group topology.

Contribution

It provides a necessary and sufficient condition for complete reducibility of continuous representations of group algebras based on the compactness of the group.

Findings

Complete reducibility holds iff G is compact.

Nonsingular continuous representations are characterized.

Connects algebraic representation theory with topological group properties.

Abstract

It is shown that every nonsingular continuous representation of the group algebra $L^{1}(G)$ in Banach spaces is completely reducible if and only if $G$ is a compact group.