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The Connes embedding property for quantum group von Neumann algebras


Authors: Michael Brannan, Benoît Collins and Roland Vergnioux
Journal: Trans. Amer. Math. Soc. 369 (2017), 3799-3819
MSC (2010): Primary 46L65, 46L54, 20G42, 22D25
DOI: https://doi.org/10.1090/tran/6752
Published electronically: November 8, 2016
MathSciNet review: 3624393
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Abstract: For a compact quantum group $ \mathbb{G}$ of Kac type, we study the existence of a Haar trace-preserving embedding of the von Neumann algebra $ L^\infty (\mathbb{G})$ into an ultrapower of the hyperfinite II$ _1$-factor (the Connes embedding property for $ L^\infty (\mathbb{G})$). We establish a connection between the Connes embedding property for $ L^\infty (\mathbb{G})$ and the structure of certain quantum subgroups of $ \mathbb{G}$ and use this to prove that the II$ _1$-factors $ L^\infty (O_N^+)$ and $ L^\infty (U_N^+)$ associated to the free orthogonal and free unitary quantum groups have the Connes embedding property for all $ N \ge 4$. As an application, we deduce that the free entropy dimension of the standard generators of $ L^\infty (O_N^+)$ equals $ 1$ for all $ N \ge 4$. We also mention an application of our work to the problem of classifying the quantum subgroups of $ O_N^+$.


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Additional Information

Michael Brannan
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 376 Altgeld Hall, 1409 W. Green Street, Urbana, Illinois 61801
Address at time of publication: Department of Mathematics, Mailstop 3368, Texas A&M University, College Station, Texas 77843-3368
Email: mbrannan@math.tamu.edu

Benoît Collins
Affiliation: Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan – and – Département de Mathématique et Statistique, Université d’Ottawa, 585 King Edward, Ottawa, Ontario K1N6N5, Canada – and – CNRS, Institut Camille Jordan, Université Lyon 1, 69622 Villeurbanne cedex, France
Email: collins@math.kyoto-u.ac.jp

Roland Vergnioux
Affiliation: UFR Sciences, LMNO, Université de Caen Basse-Normandie, Esplanade de la Paix, CS 14032, 14032 Caen cedex 5, France
Email: roland.vergnioux@unicaen.fr

DOI: https://doi.org/10.1090/tran/6752
Keywords: Connes embedding problem, quantum group, matricial microstates, von Neumann algebra, free entropy.
Received by editor(s): January 7, 2015
Received by editor(s) in revised form: May 14, 2015
Published electronically: November 8, 2016
Additional Notes: The first author’s research was partially supported by an NSERC postdoctoral fellowship
The second author’s research was partially supported by NSERC, ERA, Kakenhi and ANR-14-CE25-0003 funding
Article copyright: © Copyright 2016 American Mathematical Society