De Finetti Theorems for the Unitary Dual Group

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 of a de Finetti theorem similar to the case of the dual group action.