Quasinormalizers in crossed products of von Neumann algebras

TL;DR

This paper explores the connection between group actions on von Neumann algebras and the structure of their crossed products, introducing quasinormalizers as a key tool to understand properties like the Haagerup Approximation Property and almost periodicity.

Contribution

It provides a new von Neumann algebraic perspective on the Furstenberg-Zimmer distal tower and extends structure theorems to noncommutative dynamical systems.

Findings

Quasinormalizers generate an algebra capturing analytical properties.

New descriptions of the Furstenberg-Zimmer distal tower.

Examples contrasting noncommutative and classical cases.

Abstract

We study the relationship between the dynamics of the action $\alpha$ of a discrete group $G$ on a von Neumann algebra $M$, and structural properties of the associated crossed product inclusion $L(G) \subseteq M \rtimes_\alpha G$, and its intermediate subalgebras. This continues a thread of research originating in classical structural results for ergodic actions of discrete, abelian groups on probability spaces. A key tool in the setting of a noncommutative dynamical system is the set of quasinormalizers for an inclusion of von Neumann algebras. We show that the von Neumann algebra generated by the quasinormalizers captures analytical properties of the inclusion $L(G) \subseteq M \rtimes_\alpha G$ such as the Haagerup Approximation Property, and is essential to capturing "almost periodic" behavior in the underlying dynamical system. Our von Neumann algebraic point of view yields a new description of the Furstenberg-Zimmer distal tower for an ergodic action on a probability space, and we establish new versions of the Furstenberg-Zimmer structure theorems for general, tracial $W^*$-dynamical systems. We present a number of examples contrasting the noncommutative and classical settings which also build on previous work concerning singular inclusions of finite von Neumann algebras.