Stable finiteness of ample groupoid algebras, traces and applications

TL;DR

This paper investigates stable finiteness and traces in ample groupoid algebras, connecting algebraic properties with $C^*$-algebra theory, and extends results to inverse semigroup and Leavitt path algebras.

Contribution

It develops a theory of faithful traces on ample groupoid algebras and links stable finiteness to invariant means and tracial states, including new results and generalizations.

Findings

Stable finiteness implies existence of tracial states in certain cases

Developed a trace theory for ample groupoid algebras without integrability issues

Extended stable finiteness results to more general semigroup algebras

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. It turns out that for Hausdorff ample groupoids with compact unit space, having a stably finite algebra over some commutative ring implies the existence of a tracial state on its reduced $C^*$-algebra. 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.