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.