Free independence in ultraproduct von Neumann algebras and applications

TL;DR

This paper generalizes Popa's free independence results to ultraproduct von Neumann algebras, demonstrating new structural properties and applications such as stability of QWEP under free products and new inclusions with the relative Dixmier property.

Contribution

It extends free independence results to ultraproduct von Neumann algebras with modular invariance, providing new proofs and applications in operator algebra theory.

Findings

Existence of free subalgebras in ultraproduct von Neumann algebras.

QWEP stability under free products without modular theory.

New inclusions with the relative Dixmier property.

Abstract

The main result of this paper is a generalization of Popa's free independence result for subalgebras of ultraproduct ${\rm II_1}$ factors [Po95] to the framework of ultraproduct von Neumann algebras $(M^\omega, \varphi^\omega)$ where $(M, \varphi)$ is a $\sigma$-finite von Neumann algebra endowed with a faithful normal state satisfying $(M^\varphi)' \cap M = \mathbf{C} 1$. More precisely, we show that whenever $P_1, P_2 \subset M^\omega$ are von Neumann subalgebras with separable predual that are globally invariant under the modular automorphism group $(\sigma_t^{\varphi^\omega})$, there exists a unitary $v \in \mathcal U((M^\omega)^{\varphi^\omega})$ such that $P_1$ and $v P_2 v^*$ are $\ast$-free inside $M^\omega$ with respect to the ultraproduct state $\varphi^\omega$. Combining our main result with the recent work of Ando-Haagerup-Winsl\o w [AHW13], we obtain a new and direct proof, without relying on Connes-Tomita-Takesaki modular theory, that Kirchberg's quotient weak expectation property (QWEP) for von Neumann algebras is stable under free product. Finally, we obtain a new class of inclusions of von Neumann algebras with the relative Dixmier property.