Analysis and quantum groups / / Lars Tuset

• Record Nr.: UNINA9910734886203321
• Autore: Tuset Lars
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer Nature Switzerland AG, , [2022]; ©2022
• ISBN: 9783031072468; 9783031072451
• Descrizione fisica: 1 online resource (632 pages)
• Disciplina: 515.7
• Soggetti: Functional analysis; Group theory; Harmonic analysis; Grups quàntics; Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- 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.