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.
Categories: math.OA math.DS math.GR