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'''Free probability''' is a [[mathematics|mathematical]] theory which studies [[non-commutative]] [[random variable]]s. The "freeness" property is the analogue of the classical notion of [[statistical independence|independence]], and it is connected with [[free product]]s.



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'''Free probability''' is a [[mathematics|mathematical]] theory which studies [[non-commutative]] [[random variable]]s. The "freeness" or [[free independence]] property is the analogue of the classical notion of [[statistical independence|independence]], and it is connected with [[free product]]s.
This theory was initiated by [[Dan Voiculescu (mathematician)|Dan Voiculescu]] around 1986 in order to
This theory was initiated by [[Dan Voiculescu (mathematician)|Dan Voiculescu]] around 1986 in order to
attack the '''free group factors isomorphism problem''', an important unsolved problem in the theory of [[operator algebra]]s. 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 II<sub>1</sub> [[von Neumann algebra#Factors|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.
attack the '''free group factors isomorphism problem''', an important unsolved problem in the theory of [[operator algebra]]s. 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 II<sub>1</sub> [[von Neumann algebra#Factors|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.
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Later connections to [[random matrix|random matrix theory]], [[combinatorics]], [[Group representation|representations]] of [[symmetric group]]s, [[large deviations]] and other theories were established. Free probability is currently undergoing active research.
Later connections to [[random matrix|random matrix theory]], [[combinatorics]], [[Group representation|representations]] of [[symmetric group]]s, [[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 [[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 ''A''<sub>1</sub>, ..., ''A''<sub>''n''</sub> are then said to be '''free''' if the expectation of the product ''a''<sub>1</sub>...''a''<sub>''n''</sub> is zero whenever each ''a''<sub>''j''</sub> has zero expectation, lies in an ''A''<sub>''k''</sub> and no adjacent ''a''<sub>''j''</sub>'s come from the same subalgebra ''A''<sub>''k''</sub>. Random variables are free if they generate free unital subalgebras.
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 ''A''<sub>1</sub>, ..., ''A''<sub>''n''</sub> are then said to be '''freely independent''' if the expectation of the product ''a''<sub>1</sub>...''a''<sub>''n''</sub> is zero whenever each ''a''<sub>''j''</sub> has zero expectation, lies in an ''A''<sub>''k''</sub> and no adjacent ''a''<sub>''j''</sub>'s come from the same subalgebra ''A''<sub>''k''</sub>. 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 [[invariant (mathematics)|invariants]] of [[von Neumann algebra]]s 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.
One of the goals of free probability (still unaccomplished) was to construct new [[invariant (mathematics)|invariants]] of [[von Neumann algebra]]s 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.

Revision as of 06:29, 19 November 2009


Free probability is a mathematical theory which 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 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, ..., 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 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.

See also

References

  • A. Nica, R. Speicher: Lectures on the Combinatorics of Free Probability. Cambridge University Press, 2006, ISBN 0-521-85852-6
  • Fumio Hiai and Denis Petz, The Semicircle Law, Free Random Variables, and Entropy, ISBN 0-8218-2081-8
  • Voiculescu, D. V.; Dykema, K. J.; Nica, A. Free random variables. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. CRM Monograph Series, 1. American Mathematical Society, Providence, RI, 1992. ISBN 0-8218-6999-X
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