Tracially amenable actions and purely infinite crossed products

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.