Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 067, 29 pages      arXiv:2203.05852      https://doi.org/10.3842/SIGMA.2022.067
Contribution to the Special Issue on Non-Commutative Algebra, Probability and Analysis in Action

De Finetti Theorems for the Unitary Dual Group

Isabelle Baraquin a, Guillaume Cébron b, Uwe Franz a, Laura Maassen c and Moritz Weber d
a) Laboratoire de mathématiques de Besançon, UMR 6623, CNRS, Université Bourgogne Franche-Comté, 16 route de Gray, F-25000 Besançon, France
b) Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, UPS, F-31062 Toulouse, France
c) Formerly: RWTH Aachen University, Pontdriesch 10–16, 52062 Aachen, Germany
d) Saarland University, Fachbereich Mathematik, Postfach 151150, D-66041 Saarbrücken, Germany

Received March 25, 2022, in final form August 31, 2022; Published online September 13, 2022

Abstract
We prove several de Finetti theorems for the unitary dual group, also called the Brown algebra. Firstly, we provide a finite de Finetti theorem characterizing $R$-diagonal elements with an identical distribution. This is surprising, since it applies to finite sequences in contrast to the de Finetti theorems for classical and quantum groups; also, it does not involve any known independence notion. Secondly, considering infinite sequences in $W^*$-probability spaces, our characterization boils down to operator-valued free centered circular elements, as in the case of the unitary quantum group $U_n^+$. Thirdly, the above de Finetti theorems build on dual group actions, the natural action when viewing the Brown algebra as a dual group. However, we may also equip the Brown algebra with a bialgebra action, which is closer to the quantum group setting in a way. But then, we obtain a no-go de Finetti theorem: invariance under the bialgebra action of the Brown algebra yields zero sequences, in $W^*$-probability spaces. On the other hand, if we drop the assumption of faithful states in $W^*$-probability spaces, we obtain a non-trivial half a de Finetti theorem similar to the case of the dual group action.

Key words: de Finetti theorem; distributional invariance; exchangeable; Brown algebra; unitary dual group; $R$-diagonal elements; free circular elements.

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