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Concept
Free probability
Summary
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 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 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*-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, no adjacent aj's come from the same subalgebra Ak, 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 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 relation of free probability with random matrices is a key reason for the wide use of free probability in other subjects.
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