Analysis and Quantum Groups

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