191 research outputs found

    Borel reducibility and classification of von Neumann algebras

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    We announce some new results regarding the classification problem for separable von Neumann algebras. Our results are obtained by applying the notion of Borel reducibility and Hjorth's theory of turbulence to the isomorphism relation for separable von Neumann algebras

    Turbulence and Araki-Woods factors

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    Using Baire category techniques we prove that Araki-Woods factors are not classifiable by countable structures. As a result, we obtain a far reaching strengthening as well as a new proof of the well-known theorem of Woods that the isomorphism problem for ITPFI factors is not smooth. We derive as a consequence that the odometer actions of Z that preserve the measure class of a finite non-atomic product measure are not classifiable up to orbit equivalence by countable structures.Comment: 16 page

    Logic and operator algebras

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    The most recent wave of applications of logic to operator algebras is a young and rapidly developing field. This is a snapshot of the current state of the art.Comment: A minor chang

    The classification problem for automorphisms of C*-algebras

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    We present an overview of the recent developments in the study of the classification problem for automorphisms of C*-algebras from the perspective of Borel complexity theory.Comment: 21 page

    Games orbits play and obstructions to Borel reducibility

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    We introduce a new, game-theoretic approach to anti-classification results for orbit equivalence relations. Within this framework, we give a short conceptual proof of Hjorth's turbulence theorem. We also introduce a new dynamical criterion providing an obstruction to classification by orbits of CLI groups. We apply this criterion to the relation of equality of countable sets of reals, and the relations of unitary conjugacy of unitary and selfadjoint operators on the separable infinite-dimensional Hilbert space.Comment: 13 pages. Final version, to appear in Groups, Geometry, and Dynamic
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