90 research outputs found

    Examples of factors which have no Cartan subalgebras

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    We consider some conditions similar to Ozawa's condition (AO), and prove that if a non-injective factor satisfies such a condition and has the W*CBAP, then it has no Cartan subalgebras. As a corollary, we prove that II1\rm II_1 factors of universal orthogonal and unitary discrete quantum groups have no Cartan subalgebras. We also prove that continuous cores of type III1\rm III_1 factors with such a condition are semisolid as a II∞\rm II_\infty factor.Comment: 21 pages, final version, to appear in Trans. Amer. Math. So

    Free independence in ultraproduct von Neumann algebras and applications

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    The main result of this paper is a generalization of Popa's free independence result for subalgebras of ultraproduct II1{\rm II_1} factors [Po95] to the framework of ultraproduct von Neumann algebras (MΟ‰,φω)(M^\omega, \varphi^\omega) where (M,Ο†)(M, \varphi) is a Οƒ\sigma-finite von Neumann algebra endowed with a faithful normal state satisfying (MΟ†)β€²βˆ©M=C1(M^\varphi)' \cap M = \mathbf{C} 1. More precisely, we show that whenever P1,P2βŠ‚MΟ‰P_1, P_2 \subset M^\omega are von Neumann subalgebras with separable predual that are globally invariant under the modular automorphism group (Οƒtφω)(\sigma_t^{\varphi^\omega}), there exists a unitary v∈U((MΟ‰)φω)v \in \mathcal U((M^\omega)^{\varphi^\omega}) such that P1P_1 and vP2vβˆ—v P_2 v^* are βˆ—\ast-free inside MΟ‰M^\omega with respect to the ultraproduct state φω\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.Comment: 14 pages. v2: final version, to appear in J. London Math. So
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