Stable finiteness of ample groupoid algebras, traces and applications
Benjamin Steinberg
TL;DR
The work develops a comprehensive framework connecting stable finiteness of ample groupoid algebras with trace theory and invariant means. By a structural criterion based on orbit closures, discrete $K$-dense sets, and finite-stability of isotropy algebras, it provides broad stable-finiteness results for groupoid, inverse semigroup, and Leavitt path algebras, including a precise characterization for Leavitt path algebras via the no-exit condition. The paper also constructs faithful traces on complex groupoid algebras from invariant means, and shows how these traces imply stable finiteness and, in the Hausdorff case, tracial states on reduced $C^*$-algebras. An appendix extends Munn’s semigroup results to a general semigroup setting, offering semigroup-theoretic proofs and strengthening the algebraic understanding of stable finiteness. Overall, the work bridges algebraic and analytic techniques to obtain stable finiteness results with explicit structural criteria and broad applicability.
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.