"Notes on noncommutative LP and Orlicz spaces"

Identyfikator Librowy: 241362

Spis treści

Preface 7

Introduction 11

1.1. C*-algebras 17

Chapter 1. Preliminaries 17

1.2. Bounded operators 23

1.3. Von Neumann algebras 27

1.4. Unbounded operators 38

1.5. Affiliated operators 45

1.6. Generalized positive operators 47

2.1. Traces 55

Chapter 2. Noncommutative measure theory — semifinite case 55

2.2. Measurability 64

2.3. Algebraic properties of measurable operators 76

2.4. Topological properties of measurable operators 78

2.5. Order properties of measurable operators 82

2.6. Jordan morphisms on M� 87

3.1. Weights 91

Chapter 3. Weights and densities 91

3.2. Extensions of weights and traces 96

3.3. Density of weights with respect to a trace 98

4.1. Distributions and reduction to subalgebras 109

Chapter 4. A basic theory of decreasing rearrangements 109

4.2. Algebraic properties of decreasing rearrangements 118

4.3. Decreasing rearrangements and the trace 122

4.4. Integral inequalities and Monotone Convergence 131

5.1. Lp-spaces for von Neumann algebras with a trace 135

Chapter 5. Lp and Orlicz spaces for semifinite algebras 135

5.2. Introduction to Orlicz spaces 161

Chapter 6. Crossed products 187

6.1. Modular automorphism groups 188

6.2. Connes cocycle derivatives 197

6.3. Conditional expectations and operator valued weights 199

6.4. Crossed products with general group actions 201

6.5. Crossed products with abelian locally compact groups 205

6.6. Crossed products with modular automorphism groups 223

7.1. The semifinite setting revisited 237

Chapter 7. Lp and Orlicz spaces for general von Neumann algebras 237

7.2. Definition and normability of general Lp and Orlicz spaces 242

7.3. The trace functional and tr-duality for Lp-spaces 256

7.4. Dense subspaces of Lp-spaces 263

7.5. L2(M) and the standard form of a von Neumann algebra 278

Epilogue: Suggestions for further reading and study 285

Bibliography 289

Notation Index 299

Subject Index 303