Simplicity and tracial weights on non-unital reduced crossed products

TL;DR

This paper extends key theorems on simplicity and tracial weights from unital to non-unital reduced crossed products, with applications to locally compact groups and twisted crossed products, involving von Neumann algebra techniques.

Contribution

It generalizes theorems on simplicity and tracial weights to non-unital crossed products and applies these results to group extensions and twisted crossed products.

Findings

Simplicity of C*-algebras is stable under reduced crossed products over C*-simple groups.

Uniqueness of tracial weights is preserved in non-unital crossed products.

Results apply to locally compact groups and twisted crossed products, generalizing previous theorems.

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*-algebras is stable under taking reduced crossed product over discrete C*-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*-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.