Characterizing traces on crossed products of noncommutative C*-algebras

Abstract

We give complete descriptions of the tracial states on both the universal and reduced crossed products of a C*-dynamical system consisting of a unital C*-algebra and a discrete group. In particular, we also answer the question of when the tracial states are in canonical bijection with the invariant tracial states on the original C*-algebra. This generalizes the unique trace property for discrete groups. The analysis simplifies greatly in various cases, for example when the conjugacy classes of the original group are all finite, and in other cases gives previously known results, for example when the original C*-algebra is commutative. We also obtain results and examples in the case of abelian groups that contradict existing results in the literature of Bédos and Thomsen. Specifically, we give a finite-dimensional counterexample, and provide a correction to the result of Thomsen.

27 pages. This is the accepted manuscript to appear in Advances in Mathematics. It includes minor changes requested by the referee