Stable finiteness of ample groupoid algebras, traces and applications

Abstract

In this paper we study stable finiteness of ample groupoid algebras with applications to inverse semigroup algebras and Leavitt path algebras, recovering old results and proving some new ones. In addition, we develop a theory of (faithful) traces on ample groupoid algebras, mimicking the C * -algebra theory but taking advantage of the fact that our functions are simple and so do not have integrability issues, even in the non-Hausdorff setting. The theory of traces is closely connected with the theory of invariant means on Boolean inverse semigroups. We include an appendix on stable finiteness of more general semigroup algebras, improving on an earlier result of Munn, which is independent of the rest of the paper.

Finitely Presented Inverse Semigroups With Finitely Many Idempotents in Each -Class and Non-Hausdorff Universal Groupoids

“…The second author has recently initiated a study of stable finiteness of étale groupoid algebras [9] and, in particular, recovered a result of Munn showing that if S is an inverse semigroup whose D-classes have finitely many idempotents, then KS is stably finite for any field K of characteristic 0 [4]. Recall that a ring R is stably finite if M n (R) does not contain a copy of the bicyclic monoid as a subsemigroup for any n ≥ 1.…”

“…It is not difficult to show (see [9]) that if S is finitely presented and has a non-Hausdorff universal groupoid, then S cannot be written as a direct limit of inverse semigroups with Hausdorff universal groupoids. Since stable finiteness is preserved under direct limits, to really show that we cannot reduce stable finiteness of KS to the case that S has a Hausdorff universal groupoid, it is important to have a finitely presented inverse semigroup satisfying the conditions of Problem 1.1.…”