标题： 扭曲群胚的类型半群与群胚C*-代数的二分法 显示英文标题

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

摘要： 我们为任意扭曲的、不一定为Hausdorff的拓扑群胚发展了一种类型半群理论。类型半群是Cuntz半群的动力学版本。我们将它与约化和本质C*-代数的迹、理想、纯无限性和稳定有限性联系起来。如果一个扭曲群胚的约化C*-代数是单的，并且类型半群满足一种弱形式的几乎无穿透性，则该C*-代数要么是稳定有限的，要么是纯无限的。我们将我们的理论应用于Cartan包含。我们计算了与图上自相似群作用相关的可能非Hausdorff群胚的类型半群，并推导出相应Exel-Pardo代数的二分法。 显示英文摘要

摘要： 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.