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Article Dans Une Revue Symmetry, Integrability and Geometry : Methods and Applications Année : 2022

De Finetti Theorems for the unitary dual group

Résumé

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 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.

Dates et versions

hal-03616090 , version 1 (22-03-2022)

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Isabelle Baraquin, Guillaume Cébron, Uwe Franz, Laura Maassen, Moritz Weber. De Finetti Theorems for the unitary dual group. Symmetry, Integrability and Geometry : Methods and Applications, 2022, ⟨10.3842/SIGMA.2022.067⟩. ⟨hal-03616090⟩
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