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Dynamical McDuff-type properties for group actions on von Neumann algebras

Gábor Szabó, Lise Wouters

Abstract

We consider the notion of strong self-absorption for continuous actions of locally compact groups on the hyperfinite II$_1$-factor and characterize when such an action is tensorially absorbed by another given action on any separably acting von Neumann algebra. This extends the well-known McDuff property for von Neumann algebras and is analogous to the core theorems around strongly self-absorbing C$^*$-dynamics. Given a countable discrete group $G$ and an amenable action $G\curvearrowright M$ on any separably acting semi-finite von Neumann algebra, we establish a type of measurable local-to-global principle: If a given strongly self-absorbing $G$-action is suitably absorbed at the level of each fibre in the direct integral decomposition of $M$, then it is tensorially absorbed by the action on $M$. As a direct application of Ocneanu's theorem, we deduce that if $M$ has the McDuff property, then every amenable $G$-action on $M$ has the equivariant McDuff property, regardless whether $M$ is assumed to be injective or not. By employing Tomita-Takesaki theory, we can extend the latter result to the general case where $M$ is not assumed to be semi-finite.

Dynamical McDuff-type properties for group actions on von Neumann algebras

Abstract

We consider the notion of strong self-absorption for continuous actions of locally compact groups on the hyperfinite II-factor and characterize when such an action is tensorially absorbed by another given action on any separably acting von Neumann algebra. This extends the well-known McDuff property for von Neumann algebras and is analogous to the core theorems around strongly self-absorbing C-dynamics. Given a countable discrete group and an amenable action on any separably acting semi-finite von Neumann algebra, we establish a type of measurable local-to-global principle: If a given strongly self-absorbing -action is suitably absorbed at the level of each fibre in the direct integral decomposition of , then it is tensorially absorbed by the action on . As a direct application of Ocneanu's theorem, we deduce that if has the McDuff property, then every amenable -action on has the equivariant McDuff property, regardless whether is assumed to be injective or not. By employing Tomita-Takesaki theory, we can extend the latter result to the general case where is not assumed to be semi-finite.
Paper Structure (7 sections, 23 theorems, 127 equations)

This paper contains 7 sections, 23 theorems, 127 equations.

Table of Contents

  1. Preliminaries
  2. Ultrapowers of von Neumann algebras
  3. Cocycle morphisms
  4. One-sided intertwining
  5. Strongly self-absorbing actions
  6. Actions of discrete amenable groupoids
  7. Actions on arbitrary von Neumann algebras

Key Result

Theorem 1

Let $G$ be a countable discrete group and $M$ a von Neumann algebra with separable predual such that $M \cong M\bar{\otimes} \mathcal{R}$. Then every amenable action $\alpha\colon G \curvearrowright M$ is cocycle conjugate to $\alpha \otimes \mathrm{id}_\mathcal{R}\colon G\curvearrowright M\bar{\oti

Theorems & Definitions (70)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.5
  • Lemma 1.6
  • proof
  • ...and 60 more