A noncommutative de Finetti theorem: Invariance under quantum permutations is equivalent to freeness with amalgamation

Details A noncommutative de Finetti theorem: Invariance under quantum permutations is equivalent to freeness with amalgamation A noncommutative de Finetti theorem: Invariance under quantum permutations is equivalent to freeness with amalgamation

2008-07-04

arxiv.org/abs/0807.0677v1

Mathematics

Operator Algebras

17 pages

Scientific paper

10.1007/s00220-009-0802-8

We show that the classical de Finetti theorem has a canonical noncommutative counterpart if we strengthen `exchangeability' (i.e., invariance of the joint distribution of the random variables under the action of the permutation group) to invariance under the action of the quantum permutation group. More precisely, for an infinite sequence of noncommutative random variables, we prove that invariance of their joint distribution under quantum permutations is equivalent to the fact that the random variables are identically distributed and free with respect to the conditional expectation onto their tail algebra.

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