Avec cedram.org
Annales Mathématiques
Blaise Pascal
Rechercher un article
Recherche sur le site
Table des matières de ce fascicule | Article précédent | Article suivant
Amaury Freslon; Moritz Weber
On bi-free De Finetti theorems
Annales mathématiques Blaise Pascal, 23 no. 1 (2016), p. 21-51, doi: 10.5802/ambp.353
Article PDF
Class. Math.: 46L54, 46L53, 20G42
Mots clés: Quantum groups, free probability, De Finetti theorem

Résumé - 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.


[1] T. Banica. Symmetries of a generic coaction. Math. Ann., 314(4):763-780, 1999. Article |  MR 1709109 |  Zbl 0928.46038
[2] I. Charlesworth, B. Nelson and P. Skoufranis. Combinatorics of bi-freeness with amalgamation. Comm. Math. Phys., 338:801-847, 2015. Article |  MR 3351059
[3] I. Charlesworth, B. Nelson and P. Skoufranis. On two-faced families of noncommutative random variables. Canad. J. Math., 67(6):1290-1325, 2015. Article |  MR 3415654
[4] S. Curran. Quantum exchangeable sequences of algebras. Indiana Univ. Math. J., 58(3):1097-1125, 2009. Article |  MR 2541360 |  Zbl 1178.46064
[5] K.J. Dykema, C. Köstler and J.D. Williams. Quantum symmetric states on free product C*-algebras. http://arxiv.org/abs/1305.7293, 2014
[6] C. Köstler and R. Speicher. A noncommutative de Finetti theorem : invariance under quantum permutations is equivalent to freeness with amalgamation. Comm. Math. Phys., 291(2):473-490, 2009. Article |  MR 2530168 |  Zbl 1183.81099
[7] W. Liu. A noncommutative De Finetti theorem for boolean independence. J. Funct. Anal., 269(7):1950-1994, 2015. Article |  MR 3378866
[8] M. Mastnak and A. Nica. Double-ended queues and joint moments of left-right canonical operators on full Fock space. Internat. J. Math., 26(2), 2014.  MR 3319671
[9] A. Nica and R. Speicher. Lectures on the combinatorics of free probability, volume 335 of Lecture note series. London Mathematical Society, 2006  MR 2266879 |  Zbl 1133.60003
[10] R. Speicher. Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, volume 627 of Mem. Amer. Math. Soc.. AMS, 1998  MR 1407898 |  Zbl 0935.46056
[11] D.V. Voiculescu. Free probability for pairs of faces II : $2$-variables bi-free $R$-transform and systems with rank $\le 1$ commutation. http://arxiv.org/abs/1308.2035, 2013  MR 3449291
[12] D.V. Voiculescu. Free probability for pairs of faces I. Comm. Math. Phys., 332(3):955-980, 2014. Article |  MR 3262618
[13] S. Wang. Quantum symmetry groups of finite spaces. Comm. Math. Phys., 195(1):195-211, 1998. Article |  MR 1637425 |  Zbl 1013.17008
[14] S.L. Woronowicz. Compact quantum groups. Symétries quantiques (Les Houches, 1995):845-884, 1998.  MR 1616348 |  Zbl 0997.46045