Quantum independent increment processes I

David Applebaum, B.V. Rajarama Bhat, Johan Kustermans, J. Martin Lindsay, Michael Schuermann, Uwe Franz3540244069, 9783540244066

This volume is the first of two volumes containing the revised and completed notes lectures given at the school “Quantum Independent Increment Processes: Structure and Applications to Physics”. This school was held at the Alfried-Krupp-Wissenschaftskolleg in Greifswald during the period March 9 – 22, 2003, and supported by the Volkswagen Foundation. The school gave an introduction to current research on quantum independent increment processes aimed at graduate students and non-specialists working in classical and quantum probability, operator algebras, and mathematical physics.

The present first volume contains the following lectures: “L?vy Processes in Euclidean Spaces and Groups” by David Applebaum, “Locally Compact Quantum Groups” by Johan Kustermans, “Quantum Stochastic Analysis” by J. Martin Lindsay, and “Dilations, Cocycles and Product Systems” by B.V. Rajarama Bhat.

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Table of contents :
Front Matter……Page 1
Preface
……Page 3
Contents
……Page 5
Contents Of Volume II
……Page 7
List of Contributors
……Page 9
Introduction
……Page 11
David Applebaum……Page 15
1 Introduction……Page 16
2 Lecture 1: Infinite Divisibility and Lévy Processes in Euclidean Space……Page 19
3 Lévy Processes……Page 29
4 Lecture 2: Semigroups Induced by Lévy Processes……Page 39
5 Analytic Diversions……Page 43
6 Generators of Lévy Processes……Page 47
7 Lp-Markov Semigroups and Lévy Processes……Page 52
8 Lecture 3: Analysis of Jumps……Page 56
9 Lecture 4: Stochastic Integration……Page 69
10 Lecture 5: Lévy Processes in Groups……Page 83
11 Lecture 6: Two Lévy Paths to Quantum Stochastics……Page 98
References……Page 109
Johan Kustermans……Page 113
1 Elementary C*-algebra theory……Page 116
2 Locally compact quantum groups in the C*-algebra setting……Page 126
3 Compact quantum groups……Page 129
4 Weight theory on von Neumann algebras……Page 143
5 The definition of a locally compact quantum group……Page 158
6 Examples of locally compact quantum groups……Page 171
7 Appendix : several concepts……Page 186
References……Page 190
J.Martin Lindsay……Page 195
1 Spaces and Operators……Page 197
2 QS Processes……Page 228
3 QS Integrals……Page 235
4 QS Differential Equations……Page 252
5 QS Cocycles……Page 257
6 QS Dilation……Page 267
References……Page 278
1 Dilation theory basics……Page 286
2 E0-semigroups and product systems……Page 290
3 Domination and minimality……Page 295
4 Product systems: Recent developments……Page 299
References……Page 303
Index……Page 305