De Finetti theorems for easy quantum groups

Details De Finetti theorems for easy quantum groups De Finetti theorems for easy quantum groups

2009-07-19

arxiv.org/abs/0907.3314v3

Ann. Probab. 40 (2012), 401-435

Mathematics

Operator Algebras

Expanded Remark 5.4 on the case of unbounded variables, updated references. 23 pages

Scientific paper

We study sequences of noncommutative random variables which are invariant under "quantum transformations" coming from an orthogonal quantum group satisfying the "easiness" condition axiomatized in our previous paper. For 10 easy quantum groups, we obtain de Finetti type theorems characterizing the joint distribution of any infinite, quantum invariant sequence. In particular, we give a new and unified proof of the classical results of de Finetti and Freedman for the easy groups S_n, O_n, which is based on the combinatorial theory of cumulants. We also recover the free de Finetti theorem of K\"ostler and Speicher, and the characterization of operator-valued free semicircular families due to Curran. We consider also finite sequences, and prove an approximation result in the spirit of Diaconis and Freedman.

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