De Finetti Theorems for the Unitary Dual Group

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