Essential groupoid amenability and nuclearity of groupoid C*-algebras

TL;DR

This paper introduces a new construction of the essential $C^*$-algebra for étale groupoids, proposes an amenability notion weaker than topological amenability, and applies it to algebraic actions of semigroups.

Contribution

It develops an alternative construction of the essential $C^*$-algebra and introduces a weaker essential amenability condition for étale groupoids.

Findings

Every function with dense co-support is supported on 'dangerous' arrows.

Essential amenability is strictly weaker than topological amenability.

Describes Bruce-Li algebras as exotic essential $C^*$-algebras.

Abstract

We give an alternative construction of the essential $C^*$-algebra of an \'etale groupoid, along with an ``amenability'' notion for such groupoids that is implied by the nuclearity of this essential $C^*$-algebra. In order to do this we first introduce a maximal version of the essential $C^*$-algebra, and prove that every function with dense co-support can only be supported on the set of ``dangerous'' arrows. We then introduce an essential amenability condition for a groupoid, which is (strictly) weaker than its (topological) amenability. As an application, we describe the Bruce-Li algebras arising from algebraic actions of cancellative semigroups as exotic essential $C^*$-algebras.