Finiteness and paradoxical decompositions in C*-dynamical systems

Abstract

We discuss the interplay between $K$-theoretical dynamics and the structure theory of certain \cstar-algebras arising from crossed products. In the presence of sufficiently many projections we associate to each noncommutative \cstar-system $(A,G,\alpha )$ a type semigroup $S(A,G,\alpha )$ which reflects much of the spirit of the underlying action. We characterize purely infinite as well as stably finite crossed products in terms of finiteness and infiniteness in the type semigroup. We explore the dichotomy between stable finiteness and pure infiniteness in certain classes of reduced crossed products by means of paradoxical decompositions.