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 $(M^ω, φ^ω)$ where $(M, φ)$ is a $σ$-finite von Neumann algebra endowed with a faithful normal state satisfying $(M^φ)' \cap M = \mathbf{C} 1$. More precisely, we show that whenever $P_1, P_2 \subset M^ω$ are von Neumann subalgebras with separable predual that are globally invariant under the modular automorphism group $(σ_t^{φ^ω})$, there exists a unitary $v \in \mathcal U((M^ω)^{φ^ω})$ such that $P_1$ and $v P_2 v^*$ are $\ast$-free inside $M^ω$ with respect to the ultraproduct state $φ^ω$. Combining our main result with the recent work of Ando-Haagerup-Winsløw [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.