The weak containment problem for \'etale groupoids which are strongly amenable at infinity

TL;DR

This paper proves that for étale groupoids that are strongly amenable at infinity, the coincidence of their full and reduced $C^*$-algebras implies their amenability, linking algebraic properties to groupoid structure.

Contribution

It establishes a new criterion for amenability of étale groupoids based on the equality of full and reduced $C^*$-algebras under strong amenability at infinity.

Findings

Strongly amenable at infinity étale groupoids with coinciding $C^*$-algebras are amenable.

Provides a link between algebraic $C^*$-properties and groupoid amenability.

Enhances understanding of the weak containment problem for étale groupoids.

Abstract

We show that an \'etale groupoid which is strongly amenable at infinity is amenable whenever its full and reduced $C^*$-algebras coincide.