The Connes embedding property for quantum group von Neumann algebras

TL;DR

This paper investigates the Connes embedding property for von Neumann algebras associated with certain compact quantum groups, establishing new results for free orthogonal and unitary quantum groups and exploring implications for quantum subgroup classification.

Contribution

It proves the Connes embedding property for von Neumann algebras of free orthogonal and unitary quantum groups for all N ≥ 4, linking it to quantum subgroup structures.

Findings

Connes embedding property holds for $L^ ^ ext{infty}(O_N^+)$ and $L^ ^ extinfty(U_N^+)$ for all N ≥ 4.

The standard generators of these algebras have free entropy dimension equal to 1.

Application to classification of quantum subgroups of $O_N^+$.

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 $O_N^+$.