This volume presents a completely self-contained introduction to the elaborate theory of locally compact quantum groups, bringing the reader to the frontiers of present-day research. The exposition includes a substantial amount of material on functional analysis and operator algebras, subjects which in themselves have become increasingly important with the advent of quantum information theory. In particular, the rather unfamiliar modular theory of weights plays a crucial role in the theory, due to the presence of ‘Haar integrals’ on locally compact quantum groups, and is thus treated quite extensively
The topics covered are developed independently, and each can serve either as a separate course in its own right or as part of a broader course on locally compact quantum groups. The second part of the book covers crossed products of coactions, their relation to subfactors and other types of natural products such as cocycle bicrossed products, quantum doubles and doublecrossed products. Induced corepresentations, Galois objects and deformations of coactions by cocycles are also treated. Each section is followed by a generous supply of exercises. To complete the book, an appendix is provided on topology, measure theory and complex function theory.
Author(s): Lars Tuset
Publisher: Springer
Year: 2022
Language: English
Pages: 631
City: Cham
Preface
Contents
1 Introduction
2 Banach Spaces
2.1 Normed Spaces
2.2 Operators on Banach Spaces
2.3 Linear Functionals
2.4 Weak Topologies
2.5 Extreme Points
2.6 Fixed Point Theorems
2.7 The Eberlein-Krein-Smulian Theorems
2.8 Reflexivity and Functionals Attaining Extreme Values
2.9 Compact Operators on Banach Spaces
2.10 Complemented and Invariant Subspaces
2.11 An Approximation Property
2.12 Weakly Compact Operators
Exercises
3 Bases in Banach Spaces
3.1 Schauder Bases
3.2 Unconditional Convergence
3.3 Equivalent Bases
3.4 Dual Bases
3.5 The James Space J
Exercises
4 Operators on Hilbert Spaces
4.1 Hilbert Spaces
4.2 Fourier Transform Over the Reals
4.3 Fourier Series
4.4 Polar Decomposition of Operators on Hilbert Spaces
4.5 Compact Normal Operators
4.6 Fredholm Operators
4.7 Traceclass and Hilbert-Schmidt Operators
Exercises
5 Spectral Theory
5.1 Spectral Theory for Banach Algebras
5.2 Spectral Theory for C*-Algebras
5.3 Ideals and Hereditary Subalgebras
5.4 The Borel Spectral Theorem
5.5 Von Neumann Algebras
5.6 The σ-Weak Topology
5.7 The Kaplansky Density Theorem
5.8 Maximal Commutative Subalgebras
5.9 Unit Balls and Extremal Points in C*-Algebras
Exercises
6 States and Representations
6.1 States
6.2 The GNS-Representation
6.3 Pure States
6.4 Primitive Ideals and Prime Ideals
6.5 Postliminal C*-Algebras
6.6 Direct Limits
Exercises
7 Types of von Neumann Algebras
7.1 The Lattice of Projections
7.2 Normalcy
7.3 Center Valued Traces
7.4 Semifinite von Neumann Algebras
7.5 Classification of Factors
Exercises
8 Tensor Products
8.1 Tensor Products of C*-Algebras
8.2 Von Neumann Tensor Products
8.3 Completely Positive Maps
8.4 Hilbert Modules
Exercises
9 Unbounded Operators
9.1 Definitions and Basic Properties
9.2 The Cayley Transform
9.3 Sprectral Theory for Unbounded Operators
9.4 Generalized Convergence of Unbounded Operators
Exercises
10 Tomita-Takesaki Theory
10.1 Left and Right Hilbert Algebras
10.2 Weight Theory
10.3 Weights and Left Hilbert Algebras
10.4 Weights on C*-Algebras
10.5 The Modular Automorphism
10.6 Centralizers of Weights
10.7 Cocycle Derivatives
10.8 A Generalized Radon-Nikodym Theorem
10.9 Standard Form
10.10 Spatial Derivative
10.11 Weights and Conditional Expectations
10.12 The Extended Positive Part of a von Neumann Algebra
10.13 Operator Valued Weights
Exercises
11 Spectra and Type III Factors
11.1 The Arveson Spectrum
11.2 The Connes Spectrum
11.3 Classification of Type III Factors
Exercises
12 Quantum Groups and Duality
12.1 Hopf Algebras
12.2 Compact Quantum Groups
12.3 Locally Compact Quantum Groups
12.4 A Fundamental Involution
12.5 Density Conditions
12.6 The Coinverse
12.7 Relative Invariance
12.8 Invariance and the Modular Element
12.9 Modularity and Manageability
12.10 The Dual Quantum Group
Exercises
13 Special Cases
13.1 The Universal Quantum Group
13.2 Commutative and Cocommutative Quantum Groups
13.3 Amenability
Exercises
14 Classical Crossed Products
14.1 Crossed Products of Actions
14.2 Takesaki-Takai Duality
14.3 Landstad Theory
14.4 Examples of Crossed Products
Exercises
15 Crossed Products for Quantum Groups
15.1 Complete Left Invariance for Locally Compact Quantum Groups
15.2 Coactions and Integrability
15.3 Crossed Products of Coactions
15.4 Corepresentation Implementation of Coactions
Exercises
16 Generalized and Continuous Crossed Products
16.1 Cocycle Crossed Products
16.2 Cocycle Bicrossed Products
16.3 Continuous Coactions and Regularity
Exercises
17 Basic Construction and Quantum Groups
17.1 Basic Construction for Crossed Products of Quantum Groups
17.2 From the Basic Construction to Quantum Groups
Exercises
18 Galois Objects and Cocycle Deformations
18.1 Galois Objects
18.2 Deformation of C*-Algebras by Continuous Unitary 2-Cocycles
Exercises
19 Doublecrossed Products of Quantum Groups
19.1 Radon-Nikodym Derivatives of Weights Under Coactions
19.2 Doublecrossed Products
19.3 Morphisms of Quantum Groups and Associated Right Coactions
19.4 More on Doublecrossed Products
Exercises
20 Induction
20.1 Inducing Corepresentations Using Modular Theory
Exercises
Appendix
A.1 Set Theoretic Preliminaries
A.2 Cardinality and Bases of Vector Spaces
A.3 Topology
A.4 Nets and Induced Topologies
A.5 The Stone-Weierstrass Theorem
A.6 Measurability and Lp-Spaces
A.7 Radon Measures
A.8 Complex Measures
A.9 Product Integrals
A.10 The Haar-Measure
A.11 Holomorphic Functional Calculus
A.12 Applications to Linear Algebra and Differential Equations
A.13 The Theorems of Carleson, Runge and Phragmen-Lindelöf
Exercises
Bibliography
Index
