Free independence (original) (raw)
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. Let be a , i.e. a unital algebra over equipped with a unital linear functional . As an example, one could take, for a probability measure , Let be a family of unital subalgebras of .
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| dbo:abstract | 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 and so on. Let be a , 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 (discrete) 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 (these can be thought of as random variables in ), they are called freely independent if the algebras generated by and are freely independent. (en) |
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| rdfs:comment | 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. Let be a , i.e. a unital algebra over equipped with a unital linear functional . As an example, one could take, for a probability measure , Let be a family of unital subalgebras of . (en) |
| rdfs:label | Free independence (en) |
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