Tracial approximation and $\mathcal{Z}$-stability

Abstract

Let $A$ be a unital separable non-elementary amenable simple stably finite $C^{\ast }$-algebra such that its tracial state space has a $\sigma$-compact countable-dimensional extremal boundary. We show that $A$ is $\mathcal{Z}$-stable if and only if it has strict comparison and stable rank one. We show that this result also holds for non-unital cases (which may not be Morita equivalent to unital ones).