The Connes embedding property for quantum group von Neumann algebras

Abstract

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

Kirchberg's factorization property for discrete quantum groups

“…Problem 4.6. Just as the main results in [6,9], Theorems 4.3 and 4.4 exclude the case n = 3. However, we believe that all these results should equally hold for n = 3.…”

“…All CQGs are assumed to be of Kac type in this paper. In Section 3, we prove Theorem 3.3, the amenable generation theorem, which states that the Haar trace of a CQG is amenable if the quantum group is generated in the sense of Brannan-Collins-Vergnioux [6] by two quantum subgroups with amenable Haar traces. Finally, in Section 4, using Theorem 3.3 and results of Brannan-Collins-Vergnioux [6] and Brown-Dykema [7], we show that for n = 3, the Haar traces on U + n and O + n are amenable, which means equivalently that discrete quantum group duals U + n and O + n have Kirchberg's factorization property.…”

Topological generation and matrix models for quantum reflection groups

“…is an isomorphism. In this case, we write G = G i i∈I We will need the following alternative description of topological generation, which is almost immediate given Definition 2.9; see also [22,Proposition 3.5]. C)).…”

“…Combing all the above inequalities together, we finally obtain In particular, the generators X(N ) = {u ij } 1≤i,j≤N of L ∞ (S + N ) satisfy δ 0 (X(N )) = δ * (X(N )) = 1 for N ≥ 8. [22]. However, for O + N , even more is known: In [23] it was shown that in fact L ∞ (O + N ) is a strongly 1-bounded von Neumann algebra for all N ≥ 3.…”

Topological generation results for free unitary and orthogonal groups

“…The compact quantum group literature has recently seen considerable interest in the notion of topological generation (e.g. [9,4,2,3]). The term was coined in [9], and the concept naturally extends its classical counterpart, applicable to ordinary compact groups:…”