A short proof of Hulanicki's Theorem

TL;DR

This paper presents a simplified proof of Hulanicki's theorem, establishing that a locally compact group is amenable if and only if its left regular representation weakly contains all unitary representations, combining previously unconnected elements.

Contribution

It offers a new, streamlined proof of Hulanicki's theorem by integrating disparate parts of existing literature.

Findings

Simplified proof of Hulanicki's theorem

Clarifies the relationship between amenability and unitary representations

Unifies different approaches in the literature

Abstract

We outline a simple proof of Hulanicki's theorem, that a locally compact group is amenable if and only if the left regular representation weakly contains all unitary representations. This combines some elements of the literature which have not appeared together, before.