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

TL;DR

This paper extends the McDuff property to group actions on von Neumann algebras, providing a characterization of when such actions are tensorially absorbed, with implications for amenable actions and the structure of von Neumann algebras.

Contribution

It introduces a notion of strong self-absorption for group actions on von Neumann algebras and characterizes tensorial absorption, extending McDuff-type properties to a broader class of actions.

Findings

Characterization of strong self-absorption for group actions on von Neumann algebras.

A measurable local-to-global principle for tensorial absorption of actions.

Every amenable G-action on a McDuff von Neumann algebra has the equivariant McDuff property.

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.