Essential groupoid amenability and nuclearity of groupoid C*-algebras

We give an alternative construction of the essential $C^{\ast }$-algebra of an étale 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 $C^{\ast }$-algebras.