NON-COMMUTATIVE ANALYSIS

by Palle Jorgensen, Feng Tian

Overview

'This is a book to be read and worked with. For a beginning graduate student, this can be a valuable experience which at some points in fact leads up to recent research. For such a reader there is also historical information included and many comments aiming at an overview. It is inspiring and original how old material is combined and mixed with new material. There is always something unexpected included in each chapter, which one is thankful to see explained in this context and not only in research papers which are more difficult to access.'Mathematical Reviews ClippingsThe book features new directions in analysis, with an emphasis on Hilbert space, mathematical physics, and stochastic processes. We interpret 'non-commutative analysis' broadly to include representations of non-Abelian groups, and non-Abelian algebras; emphasis on Lie groups and operator algebras (C* algebras and von Neumann algebras.)A second theme is commutative and non-commutative harmonic analysis, spectral theory, operator theory and their applications. The list of topics includes shift invariant spaces, group action in differential geometry, and frame theory (over-complete bases) and their applications to engineering (signal processing and multiplexing), projective multi-resolutions, and free probability algebras.The book serves as an accessible introduction, offering a timeless presentation, attractive and accessible to students, both in mathematics and in neighboring fields.

Product Details

ISBN-13:	9789813202146
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	01/24/2017
Sold by:	Barnes & Noble
Format:	eBook
Pages:	564
File size:	38 MB
Note:	This product may take a few minutes to download.

Table of Contents

Foreword vii

Preface ix

Acknowledgments xv

Abbreviations, Notation, Some Core Theorems xxiii

I Introduction and Motivation 1

1 Subjects and User's Guide 3

1.1 Motivation 3

1.2 Key Themes in the Book: A Bird's-eye Preview 5

1.2.1 Operators in Hilbert Space 5

1.2.2 Multivariable Spectral Theory 7

1.2.3 Noncommutative Analysis 8

1.2.4 Probability 9

1.2.5 Other Neighboring Areas 12

1.2.6 Unitary Representations 13

1.3 Note on Cited Books and Papers 13

1.4 Reader Guide 15

1.5 A Word About the Exercises 16

1.6 List of Applications 17

1.7 Groups and Physics 18

II Topics from Functional Analysis and Operators in Hilbert Space: A Selection 21

2 Elementary Facts 23

2.1 A Sample of Topics 25

2.2 Duality 27

2.2.1 Duality and Measures 33

2.2.2 Other Spaces in Duality 36

2.3 Transfinite Induction (Zorn and All That) 37

2.4 Basics of Hilbert Space Theory 39

2.4.1 Positive Definite Functions 42

2.4.2 Orthonormal Bases 46

2.4.3 Bounded Operators in Hilbert Space 50

2.4.4 The Grain-Schmidt Process and Applications 55

2.5 Dirac's Notation 63

2.5.1 Three Norm-Completions 67

2.5.2 Connection to Quantum Mechanics 70

2.5.3 Probabilistic Interpretation of Parseval in Hilbert Space 75

2.6 The Lattice Structure of Projections 77

2.7 Multiplication Operators 83

2.A Hahn-Banach Theorems 85

2.B Banach-Limit 86

3 Unbounded Operators in Hilbert Space 89

3.1 Domain, Graph, and Adjoints 90

3.2 Characteristic Matrix 97

3.2.1 Commutants 101

3.3 Unbounded Operators Between Different Hilbert Spaces 102

3.3.1 An application to the Malliavin derivative 109

3.4 Normal Operators 111

3.5 Polar Decomposition 113

3.A Stone's Theorem 114

4 Spectral Theory 119

4.1 An Overview 120

4.2 Multiplication Operator Version 125

4.2.1 Transformation of Measures 128

4.2.2 Direct Integral Representation 131

4.2.3 Proof of Theorem 4.1 continued: 132

4.3 Projection-Valued Measure (PVM) 136

4.4 Convert M_φ to a PVM (projection-valued measure) 140

4.5 The Spectral Theorem for Compact Operators 144

4.5.1 Preliminaries 144

4.5.2 Integral operators 150

III Applications 153

5 GNS and Representations 155

5.1 Definitions and Facts: An Overview 158

5.2 The GNS Construction 163

5.3 States, Dual and Pre-dual 168

5.4 New Hilbert Spaces From "old" 174

5.4.1 GNS 174

5.4.2 Direct sum ⊕_α H_α 175

5.4.3 Hilbert-Schmidt operators (continuing the discussion in 1) 175

5.4.4 Tensor-Product H₁ ⊗ H₂ 176

5.4.5 Contractive Inclusion 176

5.4.6 Inflation (Dilation) 176

5.1.1 Quantum Information 177

5.4.1 Reflection Positivity (or renormalization) (H₊/N)^˜ 179

5.5 A Second Duality Principle: A Metric on the Set of Probability Measures 184

5.6 Abelian C^* -algebras 187

5.7 States and Representations 189

5.7.1 Normal States 196

5.7.2 A Dictionary of operator theory and quantum mechanics 197

5.8 Krein-Milman, Choquet, Decomposition of States 198

5.8.1 Noncommutative Radon-Nikodym Derivative 202

5.8.2 Examples of Disintegration 202

5.9 Examples of C^* -algebras 203

5.10 Examples of Representations 211

5.11 Beginning, of Multiplicity Theory 214

5.A The Fock-state, and Representation of CCR, Realized as Malliavin Calculus 220

6 Completely Positive Maps 223

6.1 Motivation 224

6.2 CP v.s. GNS 226

6.3 Stinespring's Theorem 228

6.4 Applications 233

6.5 Factorization 239

6.6 Eudomorphisms, Representations of O_N and Numerical Range 241

7 Brownian Motion 249

7.1 Introduction, Applications, and Context for Path-space Analysis 250

7.2 The Path Space 259

7.3 Decomposition of Brownian Motion 266

7.4 The Spectral Theorem: and Karhunen-Loève Decomposition 270

7.5 Large Matrices Revisited 271

8 Lie Groups, and their Unitary Representations 273

8.1 Motivation 276

8.2 Unitary One-Parameter Groups 280

8.3 Group - Algebra - Representations 281

8.3.1 Example - ax + b group 286

8.4 Induced Representations 288

8.1.1 Integral operators and induced representations 297

8.5 Example - Heisenberg group 302

8.5.1 ax + b group 305

8.6 Co-adjoint Orbits 306

8.6.1 Review of some Lie theory 306

8.7 Gårding Space 310

8.8 Decomposition of Representations 315

8.9 Summary of Induced Representations, the Example of d/dx 318

8.9.1 Equivalence and imprimitivity for induced representations 319

8.10 Connections to Nelson's Spectral Theory 322

8.11 Multiplicity Revisited 326

8.A The Stone-von Neumann Uniqueness Theorem 329

9 The Kadison-Singer Problem 333

9.1 Statement of the Problem 334

9.2 The Dixmier Trace 340

9.3 Frames in Hilbert Space 341

IV Extension of Operators 347

10 Selfadjoint Extensions 349

10.1 Extensions of Hermitian Operators 350

10.2 Cayley Transform 362

10.3 Boundary Triple 364

10.4 The Friedrichs Extension 371

10.5 Rigged Hilbert Space 376

11 Unbounded Graph-Laplacians 385

11.1 Basic Setting 387

11.1.1 Infinite Path Space 389

11.2 The Energy Hilbert Spaces H_E 389

11.3 The Graph-Laplacian 392

11.4 The Friedrichs Extension of Δ the Graph Laplacian 394

11.5 A 1D Example 395

12 Reproducing Kernel Hilbert Space 403

12.1 Fundamentals 403

12.2 Application to Optimization 407

12.2.1 Application: Least square-optimization 409

12.3 A Digression: Stochastic Processes 413

12.4 Two Extension Problems 415

12.5 The Reproducing Kernel Hilbert Space H_F 416

12.6 Type I v.s. Type II Extensions 425

12.7 The Case of e^-|x|, |x| < 1 426

12.7.1 The Selfadjoint Extensions A_θ ⊃ -iD_F 427

12.7.2 The Spectra of the s.a. Extensions A_θ ⊃ -iD_F 431

V Appendix 439

A An Overview of Functional Analysis Books (Cast of Characters) 441

B Terminology from Neighboring Areas 451

Classical Wiener measure/space

Hilbert's sixth problem

Infinite-dimensional analysis

Monte Carlo (MC) simulation

Multiresolution analysis (MRA)

Quantum field theory (QFT)

Quantum Information (QI)

Quantum mechanics(QM)

Quantum probability (QP)

Signal processing (SP)

Stochastic processes (SP)

Uncertainty quantification (UQ)

Unitary representations (UR)

Wavelets

C Often Cited 459

D Prizes and Fame 475

Quotes: Index of Credits 477

E List of Exercises 479

F Definitions of Frequently Occurring Terms 485

G List of Figures 489

List of Tables 493

Bibliography 495

Index 525