De Finetti theorems for the unitary dual group (a.k.a. the Brown-Glockner-von Waldenfels algebra)

Seminar: 
Discrete harmonic and non-commutative probabilistics analysis
Speaker: 
Uwe Franz (Université de Franche-Comté)
Date: 
Thursday, 15. April 2021 - 10:30
Room: 
zoom.us (kontakt: Wiktor.Ejsmont@math.uni.wroc.pl)
Abstract: 
I will present several De Finetti theorems for the unitary dual group, also known as the Brown or Brown-Glockner-von Waldenfels algebra. This algebra, which is the universal *-algebra generated by the coefficients of a unitary, has the structure of a (*-algebraic) dual group (in the sense of Voiculescu) and also the structure of an involutive bialgebra. Therefore we can consider two different kinds of actions of this algebra on sequences of noncommutative random variables. Furthermore, the five universal notions of independence (as classified by Muraki) lead to fives different notions of invariance under the action as a dual group, which have different distributional characterizations. In particular, we show that a finite sequence of noncommutative random variables is invariant under the unitary dual group w.r.t. the free product iff it is composed of certain R-diagonal elements. Based on joint work with Isabelle Baraquin, Guillaume C\'ebron, Laura Maassen and Moritz Weber.