Crossed products of dynamical systems; rigidity Vs. strong proximality

Given a dynamical system (X, Γ), the corresponding crossed product C^*-algebra C(X) ᵣ Γ is called reflecting, when every intermediate C^*-algebra C^*ᵣ(Γ) < A < C(X) ᵣ Γ is of the form A = C(Y) ᵣ Γ, 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 Γ be a non-elementary convergence group or a lattice in SLd(R) for some d ≥ 2. We show that any uniformly rigid system (X,Γ) is almost reflecting. In particular, this holds for any equicontinuous action. In the von Neumann setting, for the same groups Γ and any uniformly rigid system (X,B,μ, Γ) the crossed product algebra L∞(X,μ) Γ 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.