Akshara Prasad | Searcharxiv

arXiv2026

Topologically free minimal actions without dynamical comparison

We show the existence of a topologically free minimal action of $\mathbb F_\infty$ on the Cantor space that does not have dynamical comparison. Moreover, we show that this phenomenon can happen both in the presence and in the absence of invariant measures. We also show that strict comparison of the reduced crossed product C*-algebra does not imply dynamical comparison for minimal actions. Our technique involves constructing a monoid which is not almost unperforated, embedding it into a countable refinement monoid and then realising it as the type semigroup associated to a dynamical system.

math.DS↗

arXiv2025

Type semigroups for twisted groupoids and a dichotomy for groupoid C*-algebras

We develop a theory of type semigroups for arbitrary twisted, not necessarily Hausdorff \'etale 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.

math.OA↗