The tracial Rokhlin property for actions of amenable groups on $C^*$-algebras

Abstract

In this paper, we present a definition of the tracial Rokhlin property for (cocyclic) actions of countable discrete amenable groups on simple $C^*$-algebras, which generalize Matui and Sato's definition. We show that generic examples, like Bernoulli shift on the tensor product of copies of the Jiang-Su algebra, has the weak tracial Rokhlin property, while it is shown in {Hirshberg 2014} that such an action does not have finite Rokhlin dimension. We further show that forming a crossed product from actions with the tracial Rokhlin property preserves the class of $C^*$-algebras with real rank $0$, stable rank $1$ and has strict comparison for pro\-jec\-tions, generalizing the structural results in {Osaka 2006}. We use the same idea of the proof with significant simplification. In another joint paper with Chris Phillips and Joav Orovitz, we shall show that pureness and $\mathcal{Z} $-stability could be preserved by crossed product of actions with the weak tracial Rokhlin property. The combination of these results yields an application to the classification program, which is discussed in the aforementioned paper. These results indicate that we have the correct definition of tracial Rokhlin property for actions of general countable discrete amenable groups.