Free independence
In the mathematical theory of free probability, 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 is played by the notion of a free product of 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 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 and a state on. A final example is the group algebra of a group 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 , they are called
freely independent if the algebras generated by and are freely independent.Examples of free independence
- Let be the free product 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. Then become asymptotically freely independent as..
- More generally, independent random matrices tend to be asymptotically freely independent, under certain conditions.
