Amenable actions on finite simple C*-algebras arising from flows on Pimsner algebras

Abstract

Associated to a family of $G$-$\ast$-endomorphisms on a $G$-C*-algebra $A$ satisfying certain minimality conditions, we give a $G$-C*-correspondence $\mathcal{E}$ over $A$ whose Cuntz--Pimsner algebra ${\mathcal{O}}_{\mathcal{E}}$ is simple. For certain quasi-free flows $\gamma$ (commuting with the $G$-action) on ${\mathcal{O}}_{\mathcal{E}}$, we further prove the simplicity of the reduced crossed product ${\mathcal{O}}_{\mathcal{E}}{⋊}_{\gamma }\mathbb{R}$. We then classify the KMS weights of $\gamma$. This in particular gives a sufficient condition for ${\mathcal{O}}_{\mathcal{E}}$ and ${\mathcal{O}}_{\mathcal{E}}{⋊}_{\gamma }\mathbb{R}$ to be stably finite (and to be stably projectionless). As the amenability of $G\curvearrowright A$ inherits to the induced actions $G\curvearrowright {\mathcal{O}}_{\mathcal{E}},{\mathcal{O}}_{\mathcal{E}}{⋊}_{\gamma }\mathbb{R}$, this provides a new systematic framework to provide amenable actions on stably finite simple C*-algebras.

33 pages