General de Finetti type theorems in noncommutative probability

TL;DR

This paper establishes broad de Finetti type theorems for classical, free, and Boolean independence in noncommutative probability, extending previous work to all non-easy quantum groups and defining new Boolean quantum semigroups.

Contribution

It generalizes de Finetti theorems to all non-easy quantum groups and introduces Boolean quantum semigroups with a corresponding de Finetti theorem.

Findings

De Finetti theorems hold for all non-easy quantum groups.

Maximal distributional symmetries are identified.

A de Finetti theorem for Boolean independence is established.

Abstract

We prove general de Finetti type theorems for classical and free independence. The de Finetti type theorems work for all non-easy quantum groups, which generalize a recent work of Banica, Curran and Speicher. We determine maximal distributional symmetries which means the corresponding de Finetti type theorem fails if a sequence of random variables satisfy more symmetry relations other than the maximal one. In addition, we define Boolean quantum semigroups in analogous to the easy quantum groups, by universal conditions on matrix coordinate generators and an orthogonal projection. Then, we show a general de Finetti type theorem for Boolean independence.