We prove simplicity of all intermediate $C^*$-algebras $C^*_{ r}(\Gamma)\subseteq \mathcal{ B} \subseteq \Gamma\ltimes_r C(X)$ in the case of minimal actions of $C^*$-simple groups $\Gamma$ on compact spaces $X$. For this, we use the notion of stationary states, recently introduced by Hartman and Kalantar (arXiv:1712.10133v2). We show that the Powers' averaging property holds for the reduced crossed product $\Gamma\ltimes_r \mathcal{ A}$ for any action $\Gamma\curvearrowright \mathcal{ A}$ of a $C^*$-simple group $\Gamma$ on a unital $C^*$-algebra $\mathcal{ A}$, and use it to prove a one-to-one correspondence between stationary states on $\mathcal{ A}$ and those on $\Gamma\ltimes_r \mathcal{ A}$.
Categories: math.OA