Crossed products of dynamical systems; rigidity Vs. strong proximality

Given a dynamical system (X, ), the corresponding crossed product C^*-algebra C (X) ₑ is called reflecting, when every intermediate C^*-algebra C^*ᵣ () < A < C (X) ₑ is of the form A = C (Y) ₑ, corresponding to a dynamical factor X Y. It is called almost reflecting if E (A) A for every such A. These two notions coincide for groups admitting the approximation property (AP). Let be a non-elementary convergence group or a lattice in SLd (R) for some d 2. We show that any uniformly rigid system (X, ) is almost reflecting. In particular, this holds for any equicontinuous action. In the von Neumann setting, for the same groups and any uniformly rigid system (X, B, , ) the crossed product algebra L^ (X, ) is reflecting. An inclusion of algebras A B is called minimal ambient if there are no intermediate algebras. As a demonstration of our methods, we construct examples of minimal ambient inclusions with various interesting properties in the C^* and the von Neumann settings.