The Baum–Connes conjecture for free orthogonal quantum groups

Abstract

We prove an analogue of the Baum-Connes conjecture for free orthogonal quantum groups. More precisely, we show that these quantum groups have a γ-element and that γ = 1. It follows that free orthogonal quantum groups are K-amenable. We compute explicitly their K-theory and deduce in the unimodular case that the corresponding reduced C * -algebras do not contain nontrivial idempotents. Our approach is based on the reformulation of the Baum-Connes conjecture by Meyer and Nest using the language of triangulated categories. An important ingredient is the theory of monoidal equivalence of compact quantum groups developed by Bichon, De Rijdt and Vaes. This allows us to study the problem in terms of the quantum group SUq(2). The crucial part of the argument is a detailed analysis of the equivariant Kasparov theory of the standard Podleś sphere.2000 Mathematics Subject Classification. 20G42, 46L80, 19K35.

The Haagerup property for locally compact quantum groups

“…O G/, where O G is the dual quantum group of G. Thus, by analogy with the fact that the Haagerup property is a von Neumann property of a discrete group [20], Brannan's result can be viewed as the statement that the discrete dual quantum groups of the free orthogonal and unitary quantum groups have the Haagerup property. It was shown in [75] that the duals of free orthogonal groups satisfy an appropriate version of the Baum-Connes conjecture.…”

The $$K$$ -theory of free quantum groups

“…As a consequence we conclude in particular that the reduced C * -algebras of unimodular free quantum groups do not contain nontrivial idempotents, extending the results of Pimsner and Voiculescu for free groups mentioned in the beginning. This paper can be viewed as a continuation of [33], where the Baum-Connes conjecture for free orthogonal quantum groups was studied. Our results here rely on the work in [33] on the one hand, and on geometric arguments using actions on quantum trees in the spirit of [31] on the other hand.…”

“…This paper can be viewed as a continuation of [33], where the Baum-Connes conjecture for free orthogonal quantum groups was studied. Our results here rely on the work in [33] on the one hand, and on geometric arguments using actions on quantum trees in the spirit of [31] on the other hand. To explain the general strategy let us consider first the case of free unitary quantum groups G = FU (Q) for Q ∈ GL n (C) satisfying QQ = ±1.…”

“…It was shown by Banica [5] that FU (Q) is a quantum subgroup of the free product FO(Q) * Z in this case. Since both FO(Q) and Z satisfy the strong Baum-Connes conjecture [33], [12], it suffices to prove inheritance properties of the conjecture for free products of quantum groups and for suitable quantum subgroups. In the case of free products we adapt the construction of Kasparov and Skandalis in [14] for groups acting on buildings, and this is where certain quantum trees show up naturally.…”

“…Given a matrix Q ∈ GL n (C), the full C * -algebra of the free unitary quantum group FU (Q) is the universal C * -algebra C * f (FO(Q)) = A o (Q) for these C * -algebras, compare [5]. Following [33], we use a different notation in order to emphasize that we shall view A u (Q) and A o (Q) as the full group C * -algebras of discrete quantum groups, and not as function algebras of compact quantum groups. The approach to the K-theory of free quantum groups in this paper is based on ideas and methods originating from the Baum-Connes conjecture [6], [7].…”