A noncommutative de Finetti theorem for boolean independence
Journal of Functional Analysis, 2015, 269(7): 1950-1994.
We introduce a family of quantum semigroups and its natural coactions on noncommutative polynomials. We define three invariance conditions for the joint distribution of sequences of selfadjoint noncommutative random variables associated with these coactions. For one of the invariance conditions, we show that the joint distribution of an infinite sequence of noncommutative random variables satisfy it is equivalent to the fact that the sequence of the random variables is identically distributed and boolean independent with respect to the conditional expectation onto its tail algebra. This is a boolean analogue of de Finetti theorem on exchangeable sequences. In the end of the paper, we also discuss the other two invariance conditions which lead to some trivial results.
De Finetti theorem; Quantum semigroup; Boolean independence; Distributional symmetries