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Notes on noncommutative LP and Orlicz spaces

Stanisław Goldstein, Louis Labuschagne

EЛЕКТРОННА КНИГА

Since the pioneering work of Dixmier and Segal in the early 50's, the theory of noncommutative L^P-spaces has grown into a very refined and important theory with wide applications. Despite this fact there is as yet no self-contained peer-reviewed introduction to the most general version of this theory in print. The present work aims to fill this vacuum, in the process giving fresh impetus to the theory. The first part of the book presents: the introductory theory of von Neumann algebras - also including the slightly less common theory of generalized positive operators; the various notions of measurability, allowing the interpretation of unbounded affiliated operators as "quantum"" measurable functions, with the crucial notion of τ-measurability developed in more detail; Jordan *-morphisms (representing quantum measurable transformations) that behave well with regard to τ-measurability; and finally the different types of weights that occur naturally in the theory, before presenting a Radon-Nikodym theorem for such weights. The core, second part of the book is devoted to first developing the noncommutative theory of decreasing rearrangements, before using that technology to present the basic theory of L^P and Orlicz spaces for semifinite algebras, and then the notion of crossed product, as well as the technology underlying it, indispensable for the theory of Haagerup L^P-spaces for general von Neumann algebras. With this as a foundation, we are then finally ready to present the basic structural theory of not only Haagerup L^P-spaces, but also Orlicz spaces for general von Neumann algebras.

Preface 7

Introduction 11

Chapter 1. Preliminaries 17

1. C*-algebras 17
2. Bounded operators 23
3. Von Neumann algebras 27
4. Unbounded operators 38
5. Aﬃliated operators 45
6. Generalized positive operators 47

Chapter 2. Noncommutative measure theory — semiﬁnite case 55

1. Traces 55
2. Measurability 64
3. Algebraic properties of measurable operators 76
4. Topological properties of measurable operators 78
5. Order properties of measurable operators 82
6. Jordan morphisms on M 87

Chapter 3. Weights and densities 91

1. Weights 91
2. Extensions of weights and traces 96
3. Density of weights with respect to a trace 98

Chapter 4. A basic theory of decreasing rearrangements 109

1. Distributions and reduction to subalgebras 109
2. Algebraic properties of decreasing rearrangements 118
3. Decreasing rearrangements and the trace 122
4. Integral inequalities and Monotone Convergence 131

Chapter 5. Lp and Orlicz spaces for semiﬁnite algebras 135

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

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

7.1. The semiﬁnite setting revisited 237