Ƶ-Stability of Crossed Products by Strongly Outer Actions II

We consider a crossed product of a unital simple separable nuclear stably finite $\cal Z$-stable $C^*$-algebra $A$ by a strongly outer cocycle action of a discrete countable amenable group $\Gamma$. Under the assumption that $A$ has finitely many extremal tracial states and $\Gamma$ is elementary amenable, we show that the twisted crossed product $C^*$-algebra is $\cal Z$-stable. As an application, we also prove that all strongly outer cocycle actions of the Klein bottle group on $\cal Z$ are cocycle conjugate to each other. This is the first classification result for actions of non-abelian infinite groups on stably finite $C^*$-algebras.