Lipschitz Algebras (Second Edition)

by Nik Weaver

Overview

'The book is very well-written by one of the leading figures in the subject. It is self-contained, includes relevant recent advances and is enriched by a large number of examples and illustrations. In addition to the general bibliography, each chapter includes a section of notes, which details the authorship of the main results, and provides useful hints for further readings. Undoubtedly, this edition will be received by researchers with the same success as the first one.'
European Mathematical SocietyThis is the standard reference on algebras of Lipschitz functions, written by the leading figure in the field. The second edition includes new chapters on nonlinear Banach space geometry, differentiability in metric measure spaces, and quantum metrics. This latest material reflects the importance of spaces of Lipschitz functions in a diverse range of current research directions. Every functional analyst should have some knowledge of this subject.

Product Details

ISBN-13:	9789814740630
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	06/29/2018
Pages:	472
Product dimensions:	6.00(w) x 9.00(h) x 1.06(d)

Table of Contents

Preface v

Notation Index ix

1 Lipschitz Functions 1

1.1 Classes of metric spaces 1

1.2 Lipschitz functions 3

1.3 Sums and products of metric spaces 8

1.4 Quotients of metric spaces 14

1.5 Scalar-valued Lipschitz functions 19

1.6 Rademacher's theorem 27

1.7 Notes 34

2 Lipschitz Spaces 35

2.1 Lip and Lip₀ spaces 35

2.2 Lip versus Lip₀ 42

2.3 Composition maps 49

2.4 De Leeuw's map 55

2.5 Lipschitz gauges 59

2.6 Distortion 65

2.7 Extreme points 73

2.8 Notes 78

3 The Predual 81

3.1 The Arens-Eells space 81

3.2 Examples 86

3.3 The mass transfer problem 90

3.4 Uniqueness of the predual 95

3.5 Weak* extreme points (existence) 101

3.6 Weak* extreme points (characterization) 110

3.7 Isometries of Lip spaces 118

3.8 Isometrics of Lip₀ spaces 120

3.9 Notes 124

4 Little Lipschitz Spaces, I 127

4.1 The compact case 127

4.2 The general case 134

4.3 Duality 147

4.4 Extensions of little Lipschitz functions 152

4.5 Notes 161

5 Linearization 163

5.1 Linear surrogates 163

5.2 Lipschitz extension constants 171

5.3 Isometric embedding 177

5.4 Nonseparable counterexamples 181

5.5 Notes 185

6 Lattice Structure 187

6.1 Linear complete sublattices of Lip₀(X) 187

6.2 Complete bands of Lip₀(X) 195

6.3 Stone lattices 201

6.4 Complete distributivity 207

6.5 Embedding in cubes 210

6.6 Lipschitz lattices 216

6.7 Completely distributive Lipschitz lattices 221

6.8 Notes 227

7 Algebraic Structure 229

7.1 Order complete subalgebras of Lip₀(X) 229

7.2 Order complete ideals of Lip₀(X) 235

7.3 Spectra 238

7.4 Norm closed ideals 247

7.5 Point derivations 251

7.6 Spectral synthesis 253

7.7 Notes 259

8 Little Lipschitz Spaces, II 261

8.1 Little Lipschitz sublattices 261

8.2 Little Lipschitz bands 268

8.3 Little Lipschitz subalgebras 274

8.4 Little Lipschitz ideals 277

8.5 Hölder cubes 280

8.6 Finite dimensional Hölder spaces 286

8.7 Notes 294

9 Measurable Metrics 297

9.1 Localizable measure spaces 297

9.2 Measurable pseudometries 300

9.3 Measurable relations 305

9.4 Lipschitz functions 313

9.5 Lipschitz spaces 322

9.6 Lipschitz gauges 331

9.7 Algebra and lattice structure 336

9.8 Lipschitz lattices 348

9.9 Notes 352

10 Derivations 353

10.1 L^∞-modules 353

10.2 The derivation theorem 367

10.3 The exterior derivative 377

10.4 Examples 385

10.5 Locally controlled oscillation 391

10.6 Dirichlet forms 395

10.7 The Sierpinski gasket 403

10.8 Wiener spaces 407

10.9 Notes 409

11 Quantum Metrics 411

11.1 Quantum relations 411

11.2 Quantum metrics 419

11.3 Examples 429

11.4 Spectral Lipschitz numbers 436

11.5 Commutation Lipschitz numbers 442

11.6 Notes 446

Bibliography 447

Subject Index 453