A Noncommutative de Finetti Theorem: Invariance under Quantum Permutations is Equivalent to Freeness with Amalgamation
Abstract
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 {(x_i)_{iinmathbb{N}}} , we prove that invariance of the joint distribution of the x _{ i }’s under quantum permutations is equivalent to the fact that the x _{ i }’s are identically distributed and free with respect to the conditional expectation onto the tail algebra of the x _{ i }’s.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 October 2009
 DOI:
 10.1007/s0022000908028
 arXiv:
 arXiv:0807.0677
 Bibcode:
 2009CMaPh.291..473K
 Keywords:

 Mathematics  Operator Algebras;
 Mathematics  Probability;
 Mathematics  Quantum Algebra;
 46L54;
 46L65;
 46L53;
 60G09
 EPrint:
 17 pages