Title : A dichotomy for groupoid C*-algebras.
Abstract : Notions of paradoxical decompositions appear in the work of Hausdorff, Banach, and Tarski who showed that a discrete group satisfies the amenable/paradoxical divide. In this talk we study paradoxical phenomena in the field of operator algebras; directing our focus to C*-algebras arising from dynamical systems, graphs, and groupoids. Like Tarski, we use the type semigroup construction to move from non-paradoxicality to the existence of means or traces. These semi-groups witness the stably finite/purely infinite nature of the corresponding C*-algebras.