**Mathematical Surveys and Monographs**

Volume: 77;
2000;
376 pp;
Softcover

MSC: Primary 46;
Secondary 15; 60; 94; 81; 05

Print ISBN: 978-0-8218-4135-8

Product Code: SURV/77.S

List Price: $98.00

Individual Member Price: $78.40

**Electronic ISBN: 978-1-4704-1304-0
Product Code: SURV/77.S.E**

List Price: $98.00

Individual Member Price: $78.40

# The Semicircle Law, Free Random Variables and Entropy

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*Fumio Hiai; Dénes Petz*

The book treats free probability theory, which has been extensively developed since the early 1980s. The emphasis is put on entropy and the random matrix model approach. The volume is a unique presentation demonstrating the extensive interrelation between the topics. Wigner's theorem and its broad generalizations, such as asymptotic freeness of independent matrices, are explained in detail. Consistent throughout the book is the parallelism between the normal and semicircle laws. Voiculescu's multivariate free entropy theory is presented with full proofs and extends the results to unitary operators. Some applications to operator algebras are also given.

Based on lectures given by the authors in Hungary, Japan, and Italy, the book is a good reference for mathematicians interested in free probability theory and can serve as a text for an advanced graduate course.

#### Table of Contents

# Table of Contents

## The Semicircle Law, Free Random Variables and Entropy

- Contents vii8 free
- Preface ix10 free
- Overview 112 free
- 1 Probability Laws and Noncommutative Random Variables 1930
- 2 The Free Relation 3950
- 3 Analytic Function Theory and Infinitely Divisible Laws 91102
- 4 Random Matrices and Asymptotically Free Relation 113124
- 5 Large Deviations for Random Matrices 175186
- 6 Free Entropy of Noncommutative Random Variables 245256
- 6.1 Definition and basic properties 246257
- 6.2 Calculus for power series of noncommutative variables 253264
- 6.3 Change of variable formulas for free entropy 259270
- 6.4 Additivity of free entropy 269280
- 6.5 Free entropies of unitary and non-self adjoint random variables 275286
- 6.6 Relation between different free entropies 280291

- 7 Relation to Operator Algebras 301312
- Bibliography 357368
- Index 371382 free