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Book DetailsFields Institute CommunicationsVolume: 12; 1997; 312 ppMSC: Primary 46; Secondary 47; 05; 81;
Free probability theory is a highly noncommutative probability theory, with independence based on free products instead of tensor products. The theory models random matrices in the large \(N\) limit and operator algebra free products. It has led to a surge of new results on the von Neumann algebras of free groups.
This is a volume of papers from a workshop on Random Matrices and Operator Algebra Free Products, held at The Fields Institute for Research in the Mathematical Sciences in March 1995. Over the last few years, there has been much progress on the operator algebra and noncommutative probability sides of the subject. New links with the physics of masterfields and the combinatorics of noncrossing partitions have emerged. Moreover there is a growing free entropy theory. The idea of this workshop was to bring together people working in all these directions and from an even broader free products area where future developments might lead.ReadershipGraduate students, research mathematicians, mathematical physicists, and theoretical physicists interested in operator algebras, noncommutative probability theory or random matrix models.

Table of Contents

Chapters

Philippe Biane  Free Brownian motion, free stochastic calculus, and random matrices

Michael Douglas  Large $N$ quantum field theory and matrix models

Ken Dykema  Free products of finite dimensional and other von Neumann algebras with respect to nontracial states

Emmanuel Germain  Amalgamated free product $C^*$algebras and $KK$theory

Ian Goulden and D Jackson  Connexion coefficients for the symmetric group, free products in operator algebras, and random matrices

Uffe Haagerup  On Voiculescu’s $R$ and $S$transforms for free noncommuting random variables

Alexandru Nica and Roland Speicher  $R$diagonal pairs—A common approach to Haar unitaries and circular elements

Michael Pimsner  A class of $C^*$algebras generalizing both CuntzKrieger algebras and crossed products by ${\mathbb Z}$

Florin Radulescu  An invariant for subfactors in the von Neumann algebra of a free group

Dimitri Shlyakhtenko  Limit distributions of matrices with bosonic and fermionic entries

Dimitri Shlyakhtenko  $R$transform of certain joint distributions

Roland Speicher  On universal products

Roland Speicher and Reza Woroudi  Boolean convolution

Erling Stormer  States and shifts on infinite free products of $C$*algebras

Dan Voiculescu  The analogues of entropy and of Fisher’s information measure in free probability theory. IV: Maximum entropy and freeness

A Zee  Universal correlation in random matrix theory: A brief introduction for mathematicians


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Free probability theory is a highly noncommutative probability theory, with independence based on free products instead of tensor products. The theory models random matrices in the large \(N\) limit and operator algebra free products. It has led to a surge of new results on the von Neumann algebras of free groups.
This is a volume of papers from a workshop on Random Matrices and Operator Algebra Free Products, held at The Fields Institute for Research in the Mathematical Sciences in March 1995. Over the last few years, there has been much progress on the operator algebra and noncommutative probability sides of the subject. New links with the physics of masterfields and the combinatorics of noncrossing partitions have emerged. Moreover there is a growing free entropy theory. The idea of this workshop was to bring together people working in all these directions and from an even broader free products area where future developments might lead.
Graduate students, research mathematicians, mathematical physicists, and theoretical physicists interested in operator algebras, noncommutative probability theory or random matrix models.

Chapters

Philippe Biane  Free Brownian motion, free stochastic calculus, and random matrices

Michael Douglas  Large $N$ quantum field theory and matrix models

Ken Dykema  Free products of finite dimensional and other von Neumann algebras with respect to nontracial states

Emmanuel Germain  Amalgamated free product $C^*$algebras and $KK$theory

Ian Goulden and D Jackson  Connexion coefficients for the symmetric group, free products in operator algebras, and random matrices

Uffe Haagerup  On Voiculescu’s $R$ and $S$transforms for free noncommuting random variables

Alexandru Nica and Roland Speicher  $R$diagonal pairs—A common approach to Haar unitaries and circular elements

Michael Pimsner  A class of $C^*$algebras generalizing both CuntzKrieger algebras and crossed products by ${\mathbb Z}$

Florin Radulescu  An invariant for subfactors in the von Neumann algebra of a free group

Dimitri Shlyakhtenko  Limit distributions of matrices with bosonic and fermionic entries

Dimitri Shlyakhtenko  $R$transform of certain joint distributions

Roland Speicher  On universal products

Roland Speicher and Reza Woroudi  Boolean convolution

Erling Stormer  States and shifts on infinite free products of $C$*algebras

Dan Voiculescu  The analogues of entropy and of Fisher’s information measure in free probability theory. IV: Maximum entropy and freeness

A Zee  Universal correlation in random matrix theory: A brief introduction for mathematicians