The Connes embedding property for quantum group von Neumann algebras

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 $O_N^+$.