Cyclicity of the left regular representation of a locally compact group

TL;DR

This paper proves that the left regular representation of a locally compact group is cyclic precisely when the group is first countable, using harmonic analysis and operator algebra techniques.

Contribution

It provides a concise proof of a known result linking cyclicity of the regular representation to first countability of the group.

Findings

Left regular representation is cyclic iff the group is first countable.

Uses harmonic analysis and operator algebra methods.

Simplifies the proof of the Greenleaf and Moskowitz result.

Abstract

We combine harmonic analysis and operator algebraic techniques to give a concise argument that the left regular representation of a locally compact group is cyclic if and only if the group is first countable, a result first proved by Greenleaf and Moskowitz.