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 (corresponding to tensor products of their function algebras) is played by the notion of a free product 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 formula_1 be a non-commutative probability space, i.e. a unital algebra formula_2 over formula_3 equipped with a unital linear functional formula_4. As an example, one could take, for a probability measure formula_5,
Another example may be formula_7, the algebra of formula_8 matrices with the functional given by the normalized trace formula_9. Even more generally, formula_2 could be a von Neumann algebra and formula_11 a state on formula_2. A final example is the group algebra formula_13 of a (discrete) group formula_14 with the functional formula_11 given by the group trace formula_16.
Let formula_17 be a family of unital subalgebras of formula_2.
Definition. The family formula_17 is called "freely independent" if 
formula_20
whenever formula_21, formula_22 and formula_23.
If formula_24, formula_25 is a family of elements of formula_2 (these can be thought of as random variables in formula_2), they are called
"freely independent" if the algebras formula_28 generated by formula_29 and formula_30 are freely independent.