The Connes embedding property for quantum group von Neumann algebras
By Michael Brannan, Benoît Collins, Roland Vergnioux. 2014
53 citations recorded by the source corpus.
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 $O_N^+$.
Subfields: Atomic and Molecular Physics, and Optics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/dd53f23acf794a2d5b4376520537d581c7a203b3
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In Atomic and Molecular Physics, and Optics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Unitary Easy Quantum Groups: the free case and the group case (2015, background): “See also [BCV14] for other applications of easy quantum groups to free probability.”