Free independence in ultraproduct von Neumann algebras and applications

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.