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• Free_probability abstract "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, quantum information theory 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 C-star algebra or a von Neumann algebra. The algebra comes equipped with a noncommutative expectation, a linear functional φ: A → C such that φ(1) = 1. Unital subalgebras A1, ..., Am 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 independent unital subalgebras.One of the goals of free probability (still unaccomplished) was to construct new invariants of von Neumann algebras and free dimension is regarded as a reasonable candidate for such an invariant. The main tool used for the construction of free dimension is free entropy.The free cumulant functional (introduced by Roland Speicher) plays a major role in the theory. It is related to the lattice of noncrossing partitions of the set { 1, ..., n } in the same way in which the classic cumulant functional is related to the lattice of all partitions of that set.".
• Free_probability wikiPageExternalLink 245a-notes-5-free-probability.
• Free_probability wikiPageExternalLink comm-nas.pdf.
• Free_probability wikiPageExternalLink survey.html.
• Free_probability wikiPageExternalLink rmtool.
• Free_probability wikiPageExternalLink free.pdf.
• Free_probability wikiPageExternalLink deconvolution.pdf.
• Free_probability wikiPageID "679696".
• Free_probability wikiPageRevisionID "587797891".
• Free_probability hasPhotoCollection Free_probability.
• Free_probability subject Category:Exotic_probabilities.
• Free_probability subject Category:Free_probability_theory.
• Free_probability subject Category:Functional_analysis.
• Free_probability type Abstraction100002137.
• Free_probability type ExoticProbabilities.
• Free_probability type Measure100033615.
• Free_probability type Probability105091770.
• Free_probability comment "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.".
• Free_probability label "Free probability".
• Free_probability label "Freie Wahrscheinlichkeitstheorie".
• Free_probability sameAs Freie_Wahrscheinlichkeitstheorie.
• Free_probability sameAs m.0329d2.
• Free_probability sameAs Q515893.
• Free_probability sameAs Q515893.
• Free_probability sameAs Free_probability.
• Free_probability wasDerivedFrom Free_probability?oldid=587797891.
• Free_probability isPrimaryTopicOf Free_probability.