Tight inclusions of C*-dynamical systems

TL;DR

This paper introduces and explores the concept of tight inclusions in C*- and W*-dynamical systems, linking topological and measurable rigidity, with applications to boundary actions and von Neumann algebra injectivity.

Contribution

It defines tight inclusions of dynamical systems, analyzes their implications for boundary actions, and applies these ideas to problems of maximal injectivity in von Neumann algebras.

Findings

Tight inclusions capture boundary action rigidity.

Applications to Zimmer amenable intermediate factors.

Results on maximal injectivity of von Neumann algebras.

Abstract

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 are $C(X)\subset L^\infty(X, \nu)$ 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.