The Connes embedding property for quantum group von Neumann algebras

International audienceFor 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^+