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Home > Art, Film & Photography > Free Probability Theory: Free Probability, Free Convolution, Free Independence, Free Poisson Distribution
Free Probability Theory: Free Probability, Free Convolution, Free Independence, Free Poisson Distribution

Free Probability Theory: Free Probability, Free Convolution, Free Independence, Free Poisson Distribution


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About the Book

Purchase includes free access to book updates online and a free trial membership in the publisher's book club where you can select from more than a million books without charge. Not illustrated. Excerpt: Free probability is a mathematical theory that studies non-commutative random variables. The "freeness" or free independence property is the analogue of the classical notion of independence, and it is connected with free products. This theory was initiated by Dan Voiculescu around 1986 in order to attack the free group factors isomorphism problem, an important unsolved problem in the theory of operator algebras. Given a free group on some number of generators, we can consider the von Neumann algebra generated by the group algebra, which is a type II1 factor. The isomorphism problem asks if these are isomorphic for different numbers of generators. It is not even known if any two free group factors are isomorphic. This is similar to Tarski's free group problem, which asks whether two different non-abelian finitely generated free groups have the same elementary theory. Later connections to random matrix theory, combinatorics, representations of symmetric groups, large deviations and other theories were established. Free probability is currently undergoing active research. Typically the random variables lie in a unital algebra A such as a istar algebra or a von Neumann algebra. The algebra comes equipped with a noncommutative expectation, a linear functional: A C such that (1) = 1. Unital subalgebras A1, ..., An are then said to be freely independent if the expectation of the product a1...an is zero whenever each aj has zero expectation, lies in an Ak and no adjacent aj's come from the same subalgebra Ak. Random variables are freely independent if they generate freely indepenent unital subalgebras. One of the goals of free probability (still unaccomplished) was to construct new invariants of von Neumann algebras and free dimens... More: http: //booksllc.net/?id=679696


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Product Details
  • ISBN-13: 9781158277766
  • Publisher: Books LLC
  • Publisher Imprint: Books LLC
  • Height: 152 mm
  • Sub Title: Free Probability, Free Convolution, Free Independence, Free Poisson Distribution
  • Width: 229 mm
  • ISBN-10: 1158277768
  • Publisher Date: 19 Jun 2010
  • Binding: Paperback
  • Spine Width: 1 mm
  • Weight: 45 gr


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