Crossed products of dynamical systems; rigidity Vs. strong proximality

Abstract

Given a dynamical system $(X,\Gamma )$, the corresponding crossed product $C^{\ast }$-algebra $C(X){⋊}_r\Gamma$ is called reflecting, when every intermediate $C^{\ast }$-algebra $C_r^{\ast }(\Gamma )<\mathcal{A}<C(X){⋊}_r\Gamma$ is of the form $\mathcal{A}=C(Y){⋊}_r\Gamma$, corresponding to a dynamical factor $X\to Y$. It is called almost reflecting if $\mathbb{E}(\mathcal{A})\subset \mathcal{A}$ for every such $\mathcal{A}$. These two notions coincide for groups admitting the approximation property (AP). Let $\Gamma$ be a non-elementary convergence group or a lattice in ${\text{SL}}_d(\mathbb{R})$ for some $d\ge 2$. We show that any uniformly rigid system $(X,\Gamma )$ is almost reflecting. In particular, this holds for any equicontinuous action. In the von Neumann setting, for the same groups $\Gamma$ and any uniformly rigid system $(X,\mathcal{B},\mu ,\Gamma )$ the crossed product algebra $L^{\infty }(X,\mu )⋊\Gamma$ is reflecting. An inclusion of algebras $\mathcal{A}\subset \mathcal{B}$ is called $\text{minimal ambient}$ 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 $C^{\ast }$ and the von Neumann settings.

Minor typos corrected