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The Semicircle Law, Free Random Variables and Entropy
Fumio Hiai, Tohoku University, Aoba-ku, Sendai, Japan, and Dénes Petz, Technical University of Budapest, Hungary

Mathematical Surveys and Monographs
2000; 376 pp; softcover
Volume: 77
ISBN-10: 0-8218-4135-1
ISBN-13: 978-0-8218-4135-8
List Price: US$98
Member Price: US$78.40
Order Code: SURV/77.S
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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

  • Overview
  • Probability laws and noncommutative random variables
  • The free relation
  • Analytic function theory and infinitely divisible laws
  • Random matrices and asymptotically free relation
  • Large deviations for random matrices
  • Free entropy of noncommutative random variables
  • Relation to operator algebras
  • Bibliography
  • Index
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