Tight inclusions of $C^*$-dynamical systems

We study a notion of tight inclusions of $C^$- and $W^$-dynamical systems, which is meant to capture a tension between topological and measurable rigidity of boundary actions. An important case of such inclusions is $C(X)\subset L^\infty(X, \nu)$ for measurable boundaries with unique stationary compact models. We discuss the implications of this phenomenon in the description of Zimmer amenable intermediate factors. Furthermore, we prove applications in the problem of maximal injectivity of von Neumann algebras.