The Connes embedding property for quantum group von Neumann algebras

Michael Brannan (1) , Benoît Collins (2, 3, 4) , Roland Vergnioux (5)

1 UIUC - University of Illinois at Urbana-Champaign [Urbana]
2 Kyoto - Department of Mathematics, Kyoto University
3 Department of Mathematics and Statistics [Ottawa]
4 ICJ - Institut Camille Jordan
5 LMNO - Laboratoire de Mathématiques Nicolas Oresme

For a compact quantum group G of Kac type, we study the existence of a Haar trace-preserving embedding of the von Neumann algebra L^∞(G) into an ultrapower of the hyperfinite II_1-factor (the Connes embedding property for L^∞(G)). We establish a connection between the Connes embedding property for L^∞(G) and the structure of certain quantum subgroups of G, and use this to prove that the II_1-factors L^∞(O_N^+) and L^∞(U_N^+) associated to the free orthogonal and free unitary quantum groups have the Connes embedding property for all N >= 4. As an application, we deduce that the free entropy dimension of the standard generators of L^∞(O_N^+) equals 1 for all N >= 4. We also mention an application of our work to the problem of classifying the quantum subgroups of O_N^+.

Domaines: Algèbres d'opérateurs [math.OA]

Michael Brannan, Benoît Collins, Roland Vergnioux. The Connes embedding property for quantum group von Neumann algebras. Transactions of the American Mathematical Society, 2017, 369, pp.3799-3819. ⟨10.1090/tran/6752⟩. ⟨hal-01488929⟩