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Quasinormalizers in crossed products of von Neumann algebras

Jon Bannon, Jan Cameron, Ionut Chifan, Kunal Mukherjee, Roger Smith, Alan Wiggins

Abstract

We study the relationship between the dynamics of the action $α$ of a discrete group $G$ on a von Neumann algebra $M$, and structural properties of the associated crossed product inclusion $L(G) \subseteq M \rtimes_αG$, 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 $L(G) \subseteq M \rtimes_αG$ 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.

Quasinormalizers in crossed products of von Neumann algebras

Abstract

We study the relationship between the dynamics of the action of a discrete group on a von Neumann algebra , and structural properties of the associated crossed product inclusion , 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 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 -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.
Paper Structure (13 sections, 47 theorems, 163 equations)

This paper contains 13 sections, 47 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Background and Preliminaries
  3. $W^*$-dynamical systems
  4. Modules over finite von Neumann algebras
  5. Normalizers and quasinormalizers of $L(G) \subseteq M \rtimes_\alpha G$
  6. An action on the hyperfinite II$_1$ factor
  7. Profinite actions of i.c.c. groups
  8. Automorphism-rigid actions of discrete groups
  9. Quasinormalizers in Crossed Products and Compactness
  10. Quasinormalizers and maximal compactness
  11. Relatively almost periodic elements and compact extensions
  12. Von Neumann algebraic descriptions of the Furstenberg-Zimmer tower
  13. Approximation properties of the inclusion $L(G) \subseteq M \rtimes_\alpha G$

Key Result

Theorem A

Let $M$ be a von Neumann algebra and $\rho$ a normal, faithful state on $M$. Suppose that a discrete group $G$ acts ergodically by $\rho$-preserving automorphisms on $M$. Then

Theorems & Definitions (94)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D: CP11
  • Theorem E
  • Remark 1.1
  • Definition 2.1
  • Theorem 2.2: FaGaSm
  • Theorem 2.3
  • proof
  • ...and 84 more