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 Γ on a compact Hausdorff space X, every unital Γ−C∗-subalgebra of the reduced crossed product C(X)⋊ 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.