CONTENTS

Introduction 1

Chapter I. Preliminaries on non-crossing partitions 3

1.1. Non-crossing partitions 3

1.2. Incidence algebra and convolution 6

1.3. Multiplicative functions 7

Chapter II. Operator-valued multiplicative functions on

the lattice of non-crossing partitions 11

2.1. Operator-valued multiplicative functions 11

2.2. Connection between / and / * C 15

2.3. Special case I = I(C,G) 17

2.4. Tracial multiplicative functions 21

2.5. Product and cluster property 22

Chapter III. Amalgamated free products 26

3.1. Basic notations 26

3.2. Moment and cumulant functions 27

3.3. Definition of the amalgamated free product 32

3.4. Explicit formula for pi * (f2 35

3.5. Positivity of the amalgamated free product 42

Chapter IV. Operator-valued free probability theory 47

4.1. 5-valued random variables and free convolution 47

4.2. B-Gaussian distributions and central limit theorem 54

4.3. Positivity of .B-Gaussian distributions 56

4.4. Compound £-Poisson distributions 59

4.5. Infinitely divisible distributions 61

4.6. Full Fock space over a Hilbert-B-bimodule 63

4.7. Realization of infinitely divisible distributions on a full Fock

space 70

Chapter V. Operator-valued stochastic processes and

stochastic differential equations 74

5.1. B-valued stochastic processes 74

5.2. Formulation of the problem 77

5.3. Possible solutions of the problem 78

5.4. Gaussian approximation 83

Bibliography 85