Simplicity and tracial weights on non-unital reduced crossed products

Abstract

We extend theorems of Breuillard–Kalantar–Kennedy–Ozawa on unital reduced crossed products to the non-unital case under mild assumptions. As a result simplicity of $C^{\ast }$-algebras is stable under taking reduced crossed products over discrete $C^{\ast }$-simple groups, and a similar result for uniqueness of tracial weight. Interestingly, our analysis on tracial weights involves von Neumann algebra theory.

Our generalizations have two applications. The first is to locally compact groups. We establish stability results of (non-discrete) $C^{\ast }$-simplicity and the unique trace property under discrete group extensions. The second is to the twisted crossed product. Thanks to the Packer–Raeburn theorem, our results lead to (generalizations of) the results of Bryder–Kennedy by a different method.