# ROOM TWO: On Intermediate $C^*$-sub-algebras of Crossed Products of $C^*$-simple group actions

In this talk, we examine the structure of the intermediate $C^*$-algebras sitting between the reduced $C^*$-algebra and the reduced crossed product for $C^*$-simple group actions. We show that, for a minimal action of a $C^*$-simple group $\Gamma$ on a compact Hausdorff space $X$, every unital $\Gamma-C^*$-subalgebra of the reduced crossed product $C(X)\rtimes_{r}\Gamma$ is $\Gamma$-simple. We also show that, for a large class of actions of $C^*$-simple groups $\Gamma \curvearrowright \mathcal{A}$, including non-faithful action of any hyperbolic group, mapping class group, $\text{Out}(\mathbb{F}_n)$ or an irreducible lattice in a semisimple Lie group with trivial center and no compact factors, having a trivial amenable radical, every intermediate $C^*$-algebra $\mathcal{B}$, $C_{\lambda}^*(\Gamma)\subseteq \mathcal{B} \subseteq \mathcal{A}\rtimes_{r}\Gamma$, is of the form $\mathcal{A}_1\rtimes_{r}\Gamma$, $\mathcal{A}_1$ is a unital $\Gamma-C^*$-subalgebra of $\mathcal{A}$. A similar result holds for intermediate von Neumann algebras as well. Moreover, We shall give an example of a faithful action of a $C^*$-simple action on a unital $C^*$-algebra $\mathcal{A}$ for which the above result holds, namely the Odometer actions, leaving us with the question of whether there are other faithful actions for which such a result is true. Parts of this work are joint with Mehrdad Kalantar and Yongle Jiang.