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

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 $\mathcal{O}_\mathcal{E} \rtimes_\gamma \mathbb{R}$. We then classify the KMS weights of $\gamma$. This in particular gives a sufficient condition for $\mathcal{O}_\mathcal{E}$ and $\mathcal{O}_\mathcal{E}\rtimes_\gamma \mathbb{R}$ to be stably finite (and to be stably projectionless). As the amenability of $G \curvearrowright A$ inherits to the induced actions $G \curvearrowright \mathcal{O}_\mathcal{E}, \mathcal{O}_\mathcal{E}\rtimes_\gamma \mathbb{R}$, this provides a new systematic framework to provide amenable actions on stably finite simple C*-algebras.