POLECAMY
Koszyk
Notes on noncommutative LP and Orlicz spaces
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ibuk
Since the pioneering work of Dixmier and Segal in the early 50’s, the theory of noncommutative LP-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 LP 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 LP-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 LP-spaces, but also Orlicz spaces for general von Neumann algebras.
| Rok wydania | 2020 |
|---|---|
| Liczba stron | 308 |
| Kategoria | Publikacje darmowe |
| Wydawca | Wydawnictwo Uniwersytetu Łódzkiego |
| ISBN-13 | 978-83-8220-386-8 |
| Numer wydania | 1 |
| Język publikacji | angielski |
| Informacja o sprzedawcy | ePWN sp. z o.o. |
Ciekawe propozycje
Spis treści
| Preface | 7 |
| Introduction | 11 |
| Chapter 1. Preliminaries | 17 |
| 1.1. C*-algebras | 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 |
| Chapter 2. Noncommutative measure theory — semifinite case | 55 |
| 2.1. Traces | 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 |
| Chapter 3. Weights and densities | 91 |
| 3.1. Weights | 91 |
| 3.2. Extensions of weights and traces | 96 |
| 3.3. Density of weights with respect to a trace | 98 |
| Chapter 4. A basic theory of decreasing rearrangements | 109 |
| 4.1. Distributions and reduction to subalgebras | 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 |
| 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 |
| 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 |
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