The Baum-Connes property for compact quantum groups

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Torsion phenomena in the quantum setting

In the categorical formulation of the Baum-Connes conjecture given by R. Meyer and R. Nest for a locally compact group G we use the family F of compact subgroups of G (see Section 1.2.3). If G is discrete, then the family F is formed by the finite subgroups of G, which is exactly the torsion of G. Hence, the torsion of a discrete group allows to define an obvious complementary pair of subcategories (see Theorem 1.2.3.11), which yields the definition of the categorical assembly map. In this way, we may investigate the torsion phenomena for discrete quantum groups in order to construct the analogous complementary pair of subcategories and so the corresponding quantum assembly map. It turns out that torsion for a discrete quantum group pG can appeared under different exotic fashions. Hence, “quantum torsion” is more complicated than “classical torsion” and we don’t have yet a complete understanding of this phenomena in order to handle it in the general categorical framework of Meyer-Nest. The first notion of torsion for a discrete quantum group was introduced by R. Meyer and R. Nest [133], [131] and recently re-interpreted by Y. Arano and K. De Commer in terms of fusion rings [3].
As we shall explain in Section 1.7.2, the current formulation of the Baum-Connes conjecture for quantum groups deals only with torsion-free discrete quantum groups. Moreover, in Chapter 3 we investigate the torsion phenomena for some constructions of quantum groups (quantum direct product, quantum semi-direct product, compact bicrossed product, free product and free wreath product) in order to tackle the corresponding stability properties of the Baum-Connes property. In some cases, the torsion phenomena of a discrete quantum group can be successfully controlled in order to give a suitable Baum-Connes property formulation. For instance, this is the case for the quantum automorphism group [212] and the free wreath product Section 3.7.

Torsion à l’Arano-De Commer

The re-interpretation of torsion for quantum groups by Y. Arano and K. De Commer follows a categorical and combinatorial approach through the notion of fusions ring. Indeed, associated to any discrete quantum group we have an obvious fusion ring arising from its irreducible representations. It is advisable to keep in mind notations and definitions from Section B.3. In particular, given a compact quantum group G, we denote by ReppGq the corresponding rigid C-tensor category, which is called representation category of G. Given a subset S € IrrpGq, we denote by C : xSy the smallest full subcategory of ReppGq containing S. If, in addition, C contains the trivial representation and it is closed under taking tensor product and contragredient representations, by Tannaka-Krein-Woronowicz duality (see Theorem B.3.16 and Remark B.3.17), there is an associated C-subalgebra CpHq such that restricting the coproduct to CpHq endows it with the structure of compact quantum group H. Moreover, ReppHq naturally identifies with C and we say that pH is the quantum subgroup of pG generated by S.
Let us recall the main definitions an results about fusion rings in order to summarize the work [3] by Y. Arano and K. De Commer. We refer to [3] and [61] for more details of the subject. By the convenience of the exposition, the next presentation have been adapted with respect to the special case of fusion rings coming from discrete quantum groups (which is the relevant one for the present dissertation), so that these definitions are equivalent with the standard ones. Let pI, 1q be a pointed set with distinguished element 1, called unit of I. We equip I with an involution I ÝÑ I ÞÝÑ .

KK-theory in the quantum setting

Given a (second countable) locally compact group G, the corresponding G-equivariant Kasparov theory has been (and will be) presupposed for this dissertation and standard references for the necessary material on this subject are [86], [224], [24] or [164] (we can refer as well to the original articles of J. Cuntz, G. G. Kasparov and G. Skandalis, see for example [171], [45], [95], [97], [98]). Given a locally compact quantum group G we can construct a quantum G-equivariant Kasparov theory which imitates all the classical constructions and definitions. In this section we are going to present this quantum picture of KK-theory for the convenience of the exposition, so that it shall recall as well the classical well known Kasparov theory.
In the early work [6], S. Baaj and G. Skandalis define an equivariant KK-theory with respect to any Hopf C-algebra, extend the Kasparov product into this framework and give a particular version of the Baaj-Skandalis duality for locally compact groups and its duals. In addition, it is possible to give a more general perspective of this quantum KK-theory working with a weak Kac system (in the sense of R. Vergnioux) instead of working directly with a Hopf C-algebra. For a well detailed exposition of this we refer to Chapter 3 and Chapter 5 in [206].

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Quantum Kasparov’s theory and Baaj-Skandalis Duality

In order to simplify notations and to have a general perspective of the quantum Kasparov’s theory, we work with any Hopf C-algebra S : pS,q. However, in the context of the present dissertation such a Hopf C-algebra is supposed to be either pG pc0ppGq,pq or G pCpGq,q, where G is a compact quantum group. In order to understand the following presentation, it is advisable to keep in mind elementary notions about Hilbert modules recalled in Section A.3 and the corresponding notion of multipliers (see Definition A.4.5 and Definition A.4.6).

Compact bicrossed product

We introduce the compact bicrossed product of a matched pair of a discrete group and a compact group and we analyze some structure properties of this compact quantum group which are useful for our purpose.
It is important to say that the bicrossed product construction have had different approaches throughout the history and that we are interested in a very concrete case. More precisely, in the fundamental work [94], G. I. Kac introduced the notion of matched pair of finite groups in order to study the classification of extensions of finite groups. In the context of multiplictive unitaries, S. Baaj and G. Skandalis give in [7] a generalization of the Kac’s work defining the notion of matched pair of locally compact groups. Finally, the work of S. Vaes and L. Vainerman [196] give a very general framework for the bicrossed product defining the notion of matched pair of locally compact quantum groups. This allows in particular to develop a very technical theory by which we can give a satisfactory notion of extension of locally compact quantum groups.
If we restrict our attention to a matched pair of a discrete group and a compact group (we say compact matched pair), the resulting object is a compact quantum group and we can investigate in a much more clear fashion the properties of its representation theory and approximation properties as we can see in the work [65] due to P. Fima, K. Mukherjee and I. Patri. Actually, we refer to [65] for more details about compact matched pairs and specifically for a proof of Theorem 2.4.1 below defining the compact bicrossed product.

Table of contents :

Introduction
1 Background 
1.1 Conventions and notations
1.2 Triangulated categories
1.2.1 Elementary facts
1.2.2 Meyer-Nest’s homological algebra
1.2.3 Reformulation of the Baum-Connes conjecture
1.2.4 Meyer-Nest’s homological algebra revisited
1.3 Compact Quantum Groups
1.3.1 Woronowicz’s theory
1.3.2 Locally compact case
1.4 Actions of Quantum Groups
1.4.1 Actions of Discrete Quantum Groups
1.4.2 Spectral theory for Compact Quantum Groups
1.4.3 Induced actions from Discrete Quantum Subgroups
1.5 Crossed Products by Discrete (Quantum) Groups
1.5.1 Classical crossed products
1.5.2 Quantum crossed products
1.5.3 Further properties
1.6 Torsion phenomena in the quantum setting
1.6.1 Torsion à la Meyer-Nest
1.6.2 Torsion à l’Arano-De Commer
1.7 KK-theory in the quantum setting
1.7.1 Quantum Kasparov’s theory and Baaj-Skandalis Duality
1.7.2 Quantum Baum-Connes conjecture
2 Construction of Compact Quantum Groups 
2.1 Typical examples
2.2 Quantum direct product
2.3 Quantum semi-direct product
2.4 Compact bicrossed product
2.5 Quantum free product
2.6 Free wreath product
3 Stability properties for the QBCc 
3.1 The Baum-Connes property for compact quantum groups
3.2 The Baum-Connes property for a quantum subgroup
3.2.1 Torsion property
3.2.2 The Baum-Connes property
3.2.3 K-amenability property
3.3 The Baum-Connes property for a quantum direct product
3.3.1 Torsion property
3.3.2 The Baum-Connes property
3.3.3 K-amenability property
3.4 The Baum-Connes property for a quantum semi-direct product
3.4.1 Torsion property
3.4.2 The Baum-Connes property
3.4.3 K-amenability property
3.5 The Baum-Connes property for a compact bicrossed product
3.5.1 K-amenability property
3.6 The Baum-Connes property for a quantum free product
3.6.1 Torsion property
3.6.2 The Baum-Connes property
3.6.3 K-amenability property
3.7 The Baum-Connes property for a free wreath product
3.7.1 Torsion property
3.7.2 The Baum-Connes property
3.7.3 K-amenability property
4 An application: the K-theory for the Lemeux-Tarrago’s pHq   G {SUqp2q 
4.1 Strategies for K-theory computations
4.1.1 Torsion-free discrete quantum group case
4.1.2 Torsion discrete quantum group case
4.2 The Lemeux-Tarrago’s p Hq   G {SUqp2q
4.2.1 Preliminary computations
4.2.2 G : O􀀀pnq is a free orthogonal quantum group
4.2.3 G : U􀀀 1 . . . U􀀀 k O􀀀 1 . . . O􀀀 l is a free quantum group
4.2.4 G : Fn is the classical free group on n generators
5 Conclusion: open questions and possible lines of attack 
5.1 Stability of the Baum-Connes property
5.2 Maximal torus strategy
5.3 K-theory computations
5.4 Formulation of the Baum-Connes property for arbitrary quantum groups
Appendices
A Generalities 
A.1 Elements of C-algebras
A.2 Elements of von Neumann algebras
A.3 Elements of Hilbert modules
A.4 Elements of multiplier algebras
B Categories 
B.1 Generalities
B.2 Abelian categories
B.3 C-tensor categories. Categorical picture of Quantum Groups
Bibliography 

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