2025/02/24 by Bartosz K. Kwaśniewski, B. K. Kwaśniewski, Ralf Meyer +5
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Operator Algebra Research #Advanced Topics in Algebra #math.OA
paper · pdf · doi:10.48550/arxiv.2502.17190
openalex publication_date 2025/02/24 · openalex created_date 2025/10/06 · openalex updated_date 2026/08/01
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.