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The Semicircle Law, Free Random Variables and Entropy
 
Fumio Hiai Tohoku University, Aoba-ku, Sendai, Japan
Dénes Petz Technical University of Budapest, Budapest, Hungary
The Semicircle Law, Free Random Variables and Entropy
Softcover ISBN:  978-0-8218-4135-8
Product Code:  SURV/77.S
List Price: $129.00
MAA Member Price: $116.10
AMS Member Price: $103.20
eBook ISBN:  978-1-4704-1304-0
Product Code:  SURV/77.S.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Softcover ISBN:  978-0-8218-4135-8
eBook: ISBN:  978-1-4704-1304-0
Product Code:  SURV/77.S.B
List Price: $254.00 $191.50
MAA Member Price: $228.60 $172.35
AMS Member Price: $203.20 $153.20
The Semicircle Law, Free Random Variables and Entropy
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The Semicircle Law, Free Random Variables and Entropy
Fumio Hiai Tohoku University, Aoba-ku, Sendai, Japan
Dénes Petz Technical University of Budapest, Budapest, Hungary
Softcover ISBN:  978-0-8218-4135-8
Product Code:  SURV/77.S
List Price: $129.00
MAA Member Price: $116.10
AMS Member Price: $103.20
eBook ISBN:  978-1-4704-1304-0
Product Code:  SURV/77.S.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Softcover ISBN:  978-0-8218-4135-8
eBook ISBN:  978-1-4704-1304-0
Product Code:  SURV/77.S.B
List Price: $254.00 $191.50
MAA Member Price: $228.60 $172.35
AMS Member Price: $203.20 $153.20
  • Book Details
     
     
    Mathematical Surveys and Monographs
    Volume: 772000; 376 pp
    MSC: Primary 46; Secondary 15; 60; 94; 81; 05;

    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
     
     
    • Chapters
    • Overview
    • 1. Probability laws and noncommutative random variables
    • 2. The free relation
    • 3. Analytic function theory and infinitely divisible laws
    • 4. Random matrices and asymptotically free relation
    • 5. Large deviations for random matrices
    • 6. Free entropy of noncommutative random variables
    • 7. Relation to operator algebras
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 772000; 376 pp
MSC: Primary 46; Secondary 15; 60; 94; 81; 05;

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.

  • Chapters
  • Overview
  • 1. Probability laws and noncommutative random variables
  • 2. The free relation
  • 3. Analytic function theory and infinitely divisible laws
  • 4. Random matrices and asymptotically free relation
  • 5. Large deviations for random matrices
  • 6. Free entropy of noncommutative random variables
  • 7. Relation to operator algebras
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.