Finiteness and Paradoxical Decompostions in C*-Dynamical Systems

TL;DR

This paper explores the structure of C*-algebras from crossed products, introducing K-theoretic notions of minimality and transitivity, and characterizing their finiteness or infiniteness via associated semigroups and paradoxical decompositions.

Contribution

It introduces a K-theoretic framework for analyzing noncommutative C*-systems and characterizes pure infiniteness and stable finiteness through semigroup properties and paradoxical decompositions.

Findings

Characterization of purely infinite crossed products via semigroup properties.

Introduction of a type semigroup S(A,G) reflecting the action's nature.

Use of paradoxical decompositions to distinguish between stable finiteness and pure infiniteness.

Abstract

We discuss the interplay between K-theoretical dynamics and the structure theory for certain C*-algebras arising from crossed products. For noncommutative C*-systems we present notions of minimality and topological transitivity in the K-theoretic framework which are used to prove structural results for reduced crossed products. In the presence of sufficiently many projections we associate to each noncommutative C*-system (A,G) a type semigroup S(A,G) which reflects much of the spirit of the underlying action. We characterize purely infinite as well as stably finite crossed products by means of the infinite or rather finite nature of this semigroup. We explore the dichotomy between stable finiteness and pure infiniteness in certain classes of reduced crossed products by means of paradoxical decompositions.