Tracially amenable actions and purely infinite crossed products

TL;DR

This paper introduces tracial amenability for group actions on C*-algebras, characterizes it, and shows that such actions on certain algebras produce purely infinite crossed products, with examples involving free groups and AF-algebras.

Contribution

It defines and characterizes tracial amenability, and proves that it leads to purely infinite crossed products for actions of groups containing F_2 on specific C*-algebras.

Findings

Tracial amenability is characterized by multiple equivalent conditions.

Outer, tracially amenable actions on certain C*-algebras produce purely infinite crossed products.

Concrete examples of tracially amenable actions of free groups on AF-algebras are provided.

Abstract

We introduce the notion of tracial amenability for actions of discrete groups on unital, tracial C$^*$-algebras, as a weakening of amenability where all the relevant approximations are done in the uniform trace norm. We characterize tracial amenability with various equivalent conditions, including topological amenability of the induced action on the trace space. Our main result concerns the structure of crossed products: for groups containing the free group $F_2$, we show that outer, tracially amenable actions on simple, unital, $\mathcal{Z}$-stable C$^*$-algebras always have purely infinite crossed products. Finally, we give concrete examples of tracially amenable actions of free groups on simple, unital AF-algebras.