The Connes embedding property for quantum group von Neumann algebras

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 $O_N^{+}$.

Minor corrections and clarifications. A few bibliography updates

References in corpus (3)

• CCAP for universal discrete quantum groups
• Strong asymptotic freeness for free orthogonal quantum groups
• Examples of factors which have no Cartan subalgebras

Cited by in corpus (4)

• Bigalois extensions and the graph isomorphism game
• Orthogonal free quantum group factors are strongly 1-bounded
• Gluing compact matrix quantum groups
• Strong 1-Boundedness of Unimodular Free Orthogonal Quantum Groups