Essential groupoid amenability and nuclearity of groupoid C*-algebras
Alcides Buss, Diego Martínez
TL;DR
The paper develops a comprehensive framework connecting essential groupoid C*-algebras to amenability in non-Hausdorff étale groupoids. By introducing maximal and essential versions, along with Borel algebras and Herz–Schur multipliers, it proves that essential nuclearity is equivalent to essential (or Borel) amenability and studies the weak containment structure for both reduced and essential algebras. The results yield concrete criteria for crossed products and reveal how exotic Bruce–Li semigroup algebras arise as essential quotients, illustrating the method’s reach to non-Hausdorff and non-topologically free settings. Overall, the work provides tools to analyze nuclearity, containment, and simplicity for a broad class of groupoid C*-algebras, including non-Hausdorff and non-free actions. The approach hinges on representing noncontinuous algebraic data via essential representations and Borel multipliers, thereby extending classical amenability-nuclearity links to a wider noncommutative geometric context.
Abstract
We give an alternative construction of the essential $C^*$-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 $C^*$-algebras.