Free independence in ultraproduct von Neumann algebras and applications

نویسندگان

  • Cyril Houdayer
  • Yusuke Isono
چکیده

The main result of this paper is a generalization of Popa’s free independence result for subalgebras of ultraproduct II1 factors [Po95b] to the framework of ultraproduct von Neumann algebras (M, φ) where (M,φ) is a σ-finite von Neumann algebra endowed with a faithful normal state satisfying (M) ∩ M = C1. More precisely, we show that whenever P1, P2 ⊂ M ω are von Neumann subalgebras with separable predual that are globally invariant under the modular automorphism group (σ ω t ), there exists a unitary v ∈ U((M ) ω ) such that P1 and vP2v ∗ are ∗-free inside M with respect to the ultraproduct state φ. 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.

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عنوان ژورنال:
  • J. London Math. Society

دوره 92  شماره 

صفحات  -

تاریخ انتشار 2015