A Cocycle in the Adjoint Representation of the Orthogonal Free Quantum Groups

Abstract

Abstract. We show that the orthogonal free quantum groups are not inner amenable and we construct an explicit proper cocycle weakly contained in the regular representation. This strengthens the result of Vaes and the second author, showing that the associated von Neumann algebras are full II 1 -factors and Brannan's result showing that the orthogonal free quantum groups have Haagerup's approximation property. We also deduce Ozawa-Popa's property strong (HH) and give a new proof of Isono's result about strong solidity.

On compact bicrossed products

“…Define the sequence of dilated Chebyshev polynomials of second kind by the initial conditions P 0 (X) = 1, P 1 (X) = X and the recursion relation XP k (X) = P k+1 (X) + P k−1 (X), k ≥ 1. It is proved in [Br12] (see also [FV14]) that the net of states ω t ∈ C m (O + N ) * defined by ω t (u k ij ) = P k (t) P k (N ) δ i,j , for k ∈ Irr(O + N ) = N and t ∈ (0, 1) realize the co-Haagerup property for O + N , i.e., ω t ∈ c 0 ( O + N ) for t close to 1 and ω t → ε O + N in the weak* topology when t → 1. Now let g ∈ χ(O + N ).…”

The radial MASA in free orthogonal quantum groups

“…The argument above also proves that the C*-algebra generated by χ 1 is maximal abelian in the reduced C*-algebra C red (O + N ). From the theorem, following the strategy of [17], one can also recover the factoriality of L ∞ (O + N ) established in [20] and also in [9] (as a byproduct of non-inner amenability). (α g ⊗ id)(u n ) = (ev g ⊗ id ⊗ id)(v n 13 u n 23 ) = (1 ⊗ v n (g))u n .…”

“…• it is strongly solid [12] and has property strong HH [9], • it satisfies the Connes embedding conjecture [5].…”

“…Let us denote by S m the m-th term in brackets in (9) and note that it can only be non-zero if u m is a subrepresentation of both u k ⊗ u l and u n ⊗ u l ′ . This means that there are integers a and b such that l + k = m + 2a and n + l ′ = m + 2b.…”

Orthogonal free quantum group factors are strongly 1-bounded

“…For example, L(FO N ) is a full type II 1 -factor, it is strongly solid, and in particular prime and has no Cartan subalgebra; it has the Haagerup property (HAP), is weakly amenable with Cowling-Haagerup constant 1 (CMAP), and satisfies the Connes' Embedding conjecture [3,32,19,9,17,11,16]. Moreover, it is known that L(FO N ) behaves asymptotically like a free group factor in the sense that the canonical generators of L(FO N ) become strongly asymptotically free semicircular systems as N → ∞ [5,10].…”