The Connes embedding property for quantum group von Neumann algebras

Abstract

For a compact quantum group G \mathbb {G} of Kac type, we study the existence of a Haar trace-preserving embedding of the von Neumann algebra L ∞ ( G ) L^\infty (\mathbb {G}) into an ultrapower of the hyperfinite II 1 _1 -factor (the Connes embedding property for L ∞ ( G ) L^\infty (\mathbb {G}) ). We establish a connection between the Connes embedding property for L ∞ ( G ) L^\infty (\mathbb {G}) and the structure of certain quantum subgroups of G \mathbb {G} and use this to prove that the II 1 _1 -factors L ∞ ( O N + ) L^\infty (O_N^+) and L ∞ ( U N + ) L^\infty (U_N^+) associated to the free orthogonal and free unitary quantum groups have the Connes embedding property for all N ≥ 4 N \ge 4 . As an application, we deduce that the free entropy dimension of the standard generators of L ∞ ( O N + ) L^\infty (O_N^+) equals 1 1 for all N ≥ 4 N \ge 4 . We also mention an application of our work to the problem of classifying the quantum subgroups of O N + O_N^+ .