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.
Highlights
If H = ∗iHi is the reduced free product Hilbert space, B(Hi) can be represented on H by letting the operators act on the leftmost tensor if it is in Hi
We study possible generalizations of the de Finetti theorem to the setting of bi-freeness
This yields quantum symmetries in the sense that a family of pairs which is bi-free and identically distributed with amalgamation is invariant under the twisted action
Summary
This work is highly technical, but they were able to generalize many basic results from operator-valued free probability theory to the setting of pairs of random variables. One possible direction is to study quantum symmetries and in particular generalizations of the de Finetti theorem Speicher in [6] that an infinite family of noncommutative random variables is free and identically distributed with amalgamation over a subalgebra if and only if it is invariant under a natural action of the quantum permutation group. It is twisted so that it matches the specific combinatorics of bi-noncrossing partitions This yields quantum symmetries in the sense that a family of pairs which is bi-free and identically distributed with amalgamation is invariant under the twisted action. This gives some clues on what the combinatorics of n-freeness should be, if such a notion exists
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
