Compact bicrossed product

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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