Editing Free probability (section)

{{Short description|Mathematical theory on random variables}}
'''Free probability''' is a [[mathematics|mathematical]] theory that 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 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 of a locally compact group|group algebra]], which is a type II<sub>1</sub> [[von Neumann algebra#Factors|factor]]. The isomorphism problem asks whether 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|random matrix theory]], [[combinatorics]], [[group representation|representations]] of [[symmetric group]]s, [[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*-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>''m''</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>, no adjacent ''a''<sub>''j''</sub>'s come from the same subalgebra ''A''<sub>''k''</sub>, and ''n'' is nonzero. 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 [[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.

The relation of free probability with random matrices is a key reason for the wide use of free probability in other subjects. Voiculescu introduced the concept of freeness around 1983 in an operator algebraic context; at the beginning there was no relation at all with random matrices. This connection was only revealed later in 1991 by Voiculescu; he was motivated by the fact that the limit distribution which he found in his free central limit theorem had appeared before in Wigner's semi-circle law in the random matrix context.

The [[Cumulant#Free cumulants|free cumulant]] functional (introduced by [[Roland Speicher]])<ref name="bnt-s">{{citation
 | last = Speicher | first = Roland
 | doi = 10.1007/BF01459754
 | issue = 4
 | journal = Mathematische Annalen
 | mr = 1268597
 | pages = 611–628
 | title = Multiplicative functions on the lattice of non-crossing partitions and free convolution
 | volume = 298
 | year = 1994}}.</ref> plays a major role in the theory.  It is related to the lattice of [[noncrossing partition]]s of the set { 1, ..., ''n'' } in the same way in which the classic cumulant functional is related to the lattice of ''all'' [[partition of a set|partitions]] of that set.