De Finetti Theorems for the Unitary Dual Group

TL;DR

This paper establishes new de Finetti theorems for the unitary dual group (Brown algebra), revealing finite and infinite sequence characterizations, and explores the effects of different symmetry actions on invariance and triviality.

Contribution

It introduces novel de Finetti theorems for the Brown algebra, including finite and infinite sequence cases, and analyzes the impact of dual group and bialgebra actions.

Findings

Finite de Finetti theorem for R-diagonal elements without independence notions

Infinite sequence characterization via operator-valued free circular elements

Bialgebra invariance leads to trivial sequences unless assumptions are relaxed

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.