**Memoirs of the American Mathematical Society**

1998;
88 pp;
Softcover

MSC: Primary 46;
Secondary 60; 06

Print ISBN: 978-0-8218-0693-7

Product Code: MEMO/132/627

List Price: $48.00

Individual Member Price: $28.80

**Electronic ISBN: 978-1-4704-0216-7
Product Code: MEMO/132/627.E**

List Price: $48.00

Individual Member Price: $28.80

# Combinatorial Theory of the Free Product with Amalgamation and Operator-Valued Free Probability Theory

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*Roland Speicher*

Free probability theory, introduced by Voiculescu, has
developed very actively in the last few years and has had an
increasing impact on quite different fields in mathematics and
physics. Whereas the subject arose out of the field of von Neumann
algebras, presented here is a quite different view of Voiculescu's
amalgamated free product. This combinatorial description not only
allows re-proving of most of Voiculescu's results in a concise and
elegant way, but also opens the way for many new results.

Unlike other approaches, this book emphasizes the combinatorial
structure of the concept of “freeness”. This gives an elegant and
easily accessible description of freeness and leads to new results in
unexpected directions. Specifically, a mathematical framework for
otherwise quite ad hoc approximations in physics emerges.

#### Table of Contents

# Table of Contents

## Combinatorial Theory of the Free Product with Amalgamation and Operator-Valued Free Probability Theory

- Contents vii8 free
- Introduction 112 free
- Chapter I. Preliminaries on non-crossing partitions 314 free
- Chapter II. Operator-valued multiplicative functions on the lattice of non-crossing partitions 1122
- Chapter III. Amalgamated free products 2637
- Chapter IV. Operator-valued free probability theory 4758
- 4.1. B-valued random variables and free convolution 4758
- 4.2. B-Gaussian distributions and central limit theorem 5465
- 4.3. Positivity of B-Gaussian distributions 5667
- 4.4. Compound B-Poisson distributions 5970
- 4.5. Infinitely divisible distributions 6172
- 4.6. Full Fock space over a Hilbert-B-bimodule 6374
- 4.7. Realization of infinitely divisible distributions on a full Fock space 7081

- Chapter V. Operator-valued stochastic processes and stochastic differential equations 7485
- Bibliography 8596

#### Readership

Graduate students, research mathematicians and physicists working in functional analysis.