Abelian topological groups without irreducible Banach representations

TL;DR

This paper identifies specific abelian topological groups that lack any nontrivial strongly continuous irreducible Banach space representations, including certain Banach-Lie groups and subgroups of unitary groups.

Contribution

It demonstrates the existence of abelian topological groups without nontrivial irreducible Banach representations, expanding understanding of representation theory limitations.

Findings

Some abelian Banach-Lie groups have no nontrivial irreducible representations.

Certain monothetic subgroups of the unitary group lack such representations.

The results highlight limitations in the representation theory of specific abelian groups.

Abstract

We exhibit abelian topological groups admitting no nontrivial strongly continuous irreducible representations in Banach spaces. Among them are some abelian Banach-Lie groups and some monothetic subgroups of the unitary group of a separable Hilbert space.