In the mathematical theory of free probability

Free probability

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....

, the notion of free independence was introduced by Dan Voiculescu. The definition of free independence is parallel to the classical definition of independence, except that the role of Cartesian products of measure spaces (corresponding to tensor product

Tensor product

In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...

s of their function algebras) is played by the notion of a free product

Free product

In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties...

 of (non-commutative) probability spaces.

In the context of Voiculescu's free probability theory, many classical-probability theorems or phenomena have free probability analogs: the same theorem or phenomenon holds (perhaps with slight modifications) if the classical notion of independence is replaced by free independence. Examples of this include: the free central limit theorem; notions of free convolution; existence of free stochastic calculus and so on.

Let be a non-commutative probability space, i.e. a unital algebra over equipped with a unital linear functional . As an example, one could take, for a probability measure ,

Another example may be , the algebra of matrices with the functional given by the normalized trace . Even more generally, could be a von Neumann algebra

Von Neumann algebra

In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. They were originally introduced by John von Neumann, motivated by his study of single operators, group...

 and a state on . A final example is the group algebra

Group algebra

In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra , such that representations of the algebra are related to representations of the group...

  of a (discrete) group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

  with the functional given by the group trace .

Let be a family of unital subalgebras of .

Definition. The family is called freely independent if

whenever , and .

If , is a family of elements of (these can be thought of as random variables in ), they are called

freely independent if the algebras generated by and are freely independent.

Examples of free independence

• Let be the free product
  Free product
  In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties...
  
   of groups , let be the group algebra, be the group trace, and set . Then are freely independent.

• Let be unitary random matrices, taken independently at random from the unitary group
  Unitary group
  In mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL...
  
   (with respect to the Haar measure
  Haar measure
  In mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups....
  
  ). Then become asymptotically freely independent as . (Asymptotic freeness means that the definition of freeness holds in the limit as ).

• More generally, independent random matrices tend to be asymptotically freely independent, under certain conditions.