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Free Probability and Operator Algebras
Münster Lectures in Mathematics

Free Probability and Operator Algebras

Dan-Virgil Voiculescu (University of California, Berkeley, USA)
Nicolai Stammeier (University of Oslo, Norway)
Moritz Weber (Universität Saarbrücken, Germany)

ISBN print 978-3-03719-165-1, ISBN online 978-3-03719-665-6
DOI 10.4171/165
July 2016, 142 pages, softcover, 17 x 24 cm.
32.00 Euro

Free probability is a probability theory dealing with variables having the highest degree of noncommutativity, an aspect found in many areas (quantum mechanics, free group algebras, random matrices etc). Thirty years after its foundation, it is a well-established and very active field of mathematics. Originating from Voiculescu’s attempt to solve the free group factor problem in operator algebras, free probability has important connections with random matrix theory, combinatorics, harmonic analysis, representation theory of large groups, and wireless communication.

These lecture notes arose from a masterclass in Münster, Germany and present the state of free probability from an operator algebraic perspective. This volume includes introductory lectures on random matrices and combinatorics of free probability (Speicher), free monotone transport (Shlyakhtenko), free group factors (Dykema), free convolution (Bercovici), easy quantum groups (Weber), and a historical review with an outlook (Voiculescu). In order to make it more accessible, the exposition features a chapter on basics in free probability, and exercises for each part.

This book is aimed at master students to early career researchers familiar with basic notions and concepts from operator algebras.

Keywords: Free probability, operator algebras, random matrices, free monotone transport, free group factors, free convolution, compact quantum groups, easy quantum groups, noncrossing partitions, free independence, entropy, max-stable laws, exchangeability