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 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.