On bi-free De Finetti theorems
Abstract
We investigate possible generalizations of the de Finetti theorem to bi-free probability. We first introduce a twisted action of the quantum permutation groups corresponding to the combinatorics of bi-freeness. We then study properties of families of pairs of variables which are invariant under this action, both in the bi-noncommutative setting and in the usual noncommutative setting. We do not have a completely satisfying analogue of the de Finetti theorem, but we have partial results leading the way. We end with suggestions concerning the symmetries of a potential notion of n-freeness.
Keywords
Cite
@article{arxiv.1501.05124,
title = {On bi-free De Finetti theorems},
author = {Amaury Freslon and Moritz Weber},
journal= {arXiv preprint arXiv:1501.05124},
year = {2015}
}
Comments
16 pages. Major rewriting. In the first version the main theorem was stated through an embedding into a B-B-noncommutative probability space making it much weaker than what the proof really contains. It has therefore been split into two independent statements clarifying how far we are able to extend the de Finetti theorem to the bi-free setting