Essential groupoid amenability and nuclearity of groupoid 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.