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 $\mathrm{I}{\mathrm{I}}_1$ factors [Po95] to the framework of ultraproduct von Neumann algebras $(M^{\omega },{\phi }^{\omega })$ where $(M,\phi )$ is a $\sigma$-finite von Neumann algebra endowed with a faithful normal state satisfying $(M^{\phi }{)}^{\prime }\cap M=\mathbf{C}1$. 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 $({\sigma }_t^{{\phi }^{\omega }})$, there exists a unitary $v\in \mathcal{U}((M^{\omega }{)}^{{\phi }^{\omega }})$ such that $P_1$ and $v{P}_2{v}^{\ast }$ are $\ast$-free inside $M^{\omega }$ with respect to the ultraproduct state ${\phi }^{\omega }$. 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.

14 pages. v2: final version, to appear in J. London Math. Soc

References in corpus (2)

• Factoriality, type classification and fullness for free product von Neumann algebras
• A II${}_1$ factor approach to the Kadison-Singer problem

Cited by in corpus (1)

• Weight theory for ultraproducts