# The -theory of free quantum groups

###### Abstract.

In this paper we study the -theory of free quantum groups in the sense of Wang and Van Daele, more precisely, of
free products of free unitary and free orthogonal quantum groups. We show that these quantum groups are -amenable
and establish an analogue of the Pimsner-Voiculescu exact sequence. As a consequence, we obtain in particular an explicit
computation of the -theory of free quantum groups.

Our approach relies on a generalization of methods from the Baum-Connes conjecture to the framework of discrete quantum
groups. This is based on the categorical reformulation of the Baum-Connes conjecture developed by Meyer and Nest. As a
main result we show that free quantum groups have a -element and that .

As an important ingredient in the proof we adapt the Dirac-dual Dirac method for groups acting on trees to the
quantum case. We use this to extend some permanence properties of the Baum-Connes conjecture to our setting.

###### 2000 Mathematics Subject Classification:

20G42, 46L80, 19K35## 1. Introduction

A classical result in the theory of -algebras is the computation of the -theory of the reduced
group -algebra of the free group on generators by Pimsner and Voiculescu [PVfree].
Their result resolved in particular Kadison’s problem on the existence of nontrivial projections in
these -algebras. More generally, Pimsner and Voiculescu established an exact sequence for the -theory of
reduced crossed products by free groups [PVsequences], [PVfree]. This exact sequence is an important tool in
operator -theory.

The -theory of the full group -algebra was calculated before by Cuntz in a simple and
elegant way, based on a general formula for the -theory of free products [Cuntzfreeproduct]. Motivated by this,
Cuntz introduced the notion of -amenability for discrete groups and gave a shorter proof of the results of Pimsner and
Voiculescu [Cuntzkam]. The fact that free groups are -amenable expresses in a conceptually clear way that
full and reduced crossed products for these groups cannot be distinguished on the level of -theory.

The main aim of this paper is to obtain analogous results for free quantum groups. In fact, in the theory of discrete quantum
groups, the rôle of free quantum groups is analogous to the rôle of free groups among classical discrete groups. Roughly
speaking, any discrete quantum group can be obtained as a quotient of a free quantum group. Classically, the free group on
generators can be described as the free product

of copies of . In the quantum case there is a similar free product construction, but in contrast to the classical situation there are different building blocks out of which free quantum groups are assembled. More precisely, a free quantum group is of the form

for matrices and such that . Here
we denote by and the free unitary and free orthogonal quantum groups introduced by Wang
and Van Daele [Wangfree], [vDWuniversal]. The special case and
of this family reduces to the classical free group on generators.

In order to explain our notation let us briefly review some definitions. Given a matrix , the full
-algebra of the free unitary quantum group is the universal -algebra
generated by the entries of an -matrix satisfying the relations that and are
unitary. Here denotes the matrix obtained from by taking the transpose of its adjoint. The full -algebra
of the free orthogonal quantum group is the quotient of
by the relation . Finally, the full -algebra of the free product of two discrete quantum
groups and is the unital free product of the corresponding full
-algebras, see [Wangfree]. That is, the full -algebra of a free quantum group as above is simply the free product
of the -algebras and .

We remark that one usually writes and for
these -algebras, compare [Banicaunitary]. Following [Voigtbcfo], we use a different notation in order to emphasize
that we shall view and as the full group -algebras of discrete quantum groups, and not as
function algebras of compact quantum groups.

The approach to the -theory of free quantum groups in this paper is based on ideas and methods originating
from the Baum-Connes conjecture [BC], [BCH]. It relies in particular on the categorical
reformulation of the Baum-Connes conjecture developed by Meyer and Nest [MNtriangulated]. In fact, our
main result is that free quantum groups satisfy an analogue of the strong Baum-Connes conjecture. The precise
meaning of this statement will be explained in section LABEL:secbc, along with the necessary preparations from
the theory of triangulated categories. Together with the results of Banica on the representation theory of
free quantum groups [Banicafo], [Banicaunitary], the strong Baum-Connes property implies that every
object of the equivariant Kasparov category for a free quantum group has a projective resolution
of length one. Based on this, we immediately obtain our Pimsner-Voiculescu exact sequence by invoking some
general categorical considerations [MNhomalg1]. As a consequence we conclude in particular that the reduced -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 [Voigtbcfo], where the Baum-Connes
conjecture for free orthogonal quantum groups was studied. Our results here rely on the work in [Voigtbcfo]
on the one hand, and on geometric arguments using actions on quantum trees in the spirit of [Vergniouxtrees]
on the other hand.

To explain the general strategy let us consider first the case of free unitary quantum groups
for satisfying . It was shown by Banica [Banicaunitary]
that is a quantum subgroup of the free product in this case.
Since both and satisfy the strong Baum-Connes conjecture [Voigtbcfo],
[HKatmenable], 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 [KSbuildings] for groups acting on trees, and this is where certain quantum trees show up
naturally. An important difference to the classical situation is that one has to work equivariantly with respect
to the Drinfeld double of .

The quantum group associated to a general matrix does not admit
an embedding into a free product as above. As in [Voigtbcfo] we use an indirect argument based on the
monoidal equivalences for free quantum groups obtained by Bichon, de Rijdt, and Vaes [BdRV]. This allows
us to reduce to matrices , and in this case one may even assume
without loss of generality. We might actually restrict attention to -matrices throughout, however, this would
not simplify the arguments.

Let us now describe how the paper is organized. In section 2 we collect some preliminaries on quantum groups and fix
our notation. Section LABEL:secquantumsub contains basic facts about quantum subgroups of discrete quantum groups
and their homogeneous spaces. In section LABEL:secdivisible we introduce and discuss the notion of a divisible quantum subgroup
of a discrete quantum group. This concept appears naturally in the context of inheritance properties of the Baum-Connes conjecture.
Roughly speaking, divisible quantum subgroups are particularly well-behaved from the point of view of corepresentation theory. In
section LABEL:secdirac we define the Dirac element associated to a free product of discrete quantum groups acting on the corresponding
quantum tree. Moreover we define the dual Dirac element and show that the resulting -element is equal to the identity.
In section LABEL:secbc we review the approach to the Baum-Connes conjecture developed by Meyer and Nest. We explain in particular
the categorical ingredients needed to formulate the strong Baum-Connes property in our context. Then, using the considerations
from section LABEL:secdirac and [Voigtbcfo], we prove that free quantum groups have the strong Baum-Connes property. Finally, in
section LABEL:secktheory we discuss the main consequences of this result. As indicated above, we derive in particular the
-amenability of free quantum groups and establish an analogue of the Pimsner-Voiculescu sequence.

Let us make some remarks on notation. We write for the space of adjointable operators on a Hilbert -module .
Moreover denotes the space of compact operators. The closed linear span of a subset of a Banach space is denoted
by . Depending on the context, the symbol denotes either the tensor product of Hilbert spaces, the minimal tensor
product of -algebras, or the tensor product of von Neumann algebras. We write for algebraic tensor products. For
operators on multiple tensor products we use the leg numbering notation. If and are unital -algebras we
write for the unital free product, that is, the free product of and amalgamated over .

We would like to thank G. Skandalis for helpful comments.

## 2. Preliminaries on quantum groups

In this section we recall some basic definitions concerning quantum groups and their actions on -algebras. In
addition we review the definition of the Drinfeld double and the description of its actions in terms of the underlying
quantum group and its dual. For more information we refer to [BSUM], [Kustermansuniversal],
[KVLCQG], [Vaesimprimitivity], [Woronowiczleshouches]. Our notation and conventions will mainly follow
[NVpoincare], [Voigtbcfo].

Let be a normal, semifinite and faithful weight on a von Neumann algebra . We use the standard notation

and write for the space of positive normal linear functionals on . If is a normal unital -homomorphism, the weight is called left invariant with respect to provided

for all and . Similarly one defines right invariant weights.

###### Definition 2.1.

A locally compact quantum group is given by a von Neumann algebra together with a normal unital -homomorphism satisfying the coassociativity relation

and normal semifinite faithful weights and on which are left and right invariant, respectively.

If is a locally compact group, then the algebra of essentially bounded measurable functions on together with the comultiplication given by

defines a locally compact quantum group. The weights and are given in this case by left and right Haar measures
on , respectively. Of course, for a general locally compact quantum group the notation is purely formal.

An important tool in the study of locally compact quantum groups are multiplicative unitaries. If
is a GNS-construction for the weight , then the multiplicative unitary is the operator on given by

for all . This unitary satisfies the pentagonal equation and one can recover the von Neumann algebra as the strong closure of the algebra where denotes the space of normal linear functionals on . Moreover one has

for all . Let us remark that we will only consider quantum groups for which is a separable
Hilbert space.

The group-von Neumann algebra of the quantum group is the strong
closure of the algebra with the comultiplication
given by

where and is the flip map. It defines
a locally compact quantum group which is called the dual of . The left invariant weight
for the dual quantum group has a GNS-construction ,
and according to our conventions we have .

The reduced -algebra of functions on the quantum group is

and the reduced group -algebra of is

Moreover we have . Together with the comultiplications inherited from and , respectively, the -algebras and are Hopf--algebras in the following sense.

###### Definition 2.2.

A Hopf -algebra is a -algebra together with an injective nondegenerate -homomorphism such that the diagram