Type semigroups for twisted groupoids and a dichotomy for groupoid C*-algebras
Bartosz K. Kwaśniewski, Ralf Meyer, Akshara Prasad
TL;DR
The paper develops a dynamic analogue of the Cuntz semigroup for twisted, not-necessarily-Hausdorff étale groupoids by introducing a type semigroup $S_{}()$ built from an inverse semigroup basis. It relates this semigroup to traces, ideals, and finiteness properties of reduced and essential groupoid $C^*$-algebras, proving a robust dichotomy: simple twisted groupoid algebras are either stably finite or purely infinite, with regular states and invariant measures encoding traces. Regular ideals correspond to open invariant subsets, and regular states correspond to traces on reduced algebras, giving a precise dynamic-to-operator-algebra bridge. The framework applies to Cartan inclusions and yields a dichotomy for Exel–Pardo algebras arising from self-similar group actions on graphs, via an isomorphism $W()\cong ilde{W}() to W(E/)$, generalizing prior results for Hausdorff or amenable settings. Overall, the work extends the Cuntz semigroup paradigm to twisted groupoid dynamics, clarifying when dynamical paradoxicality forces pure infiniteness and when traces control finiteness.
Abstract
We develop a theory of type semigroups for arbitrary twisted, not necessarily Hausdorff étale groupoids. The type semigroup is a dynamical version of the Cuntz semigroup. We relate it to traces, ideals, pure infiniteness, and stable finiteness of the reduced and essential C*-algebras. If the reduced C*-algebra of a twisted groupoid is simple and the type semigroup satisfies a weak version of almost unperforation, then the C*-algebra is either stably finite or purely infinite. We apply our theory to Cartan inclusions. We calculate the type semigroup for the possibly non-Hausdorff groupoids associated to self-similar group actions on graphs and deduce a dichotomy for the resulting Exel-Pardo algebras.