Non-Commutative Stochastic Independence and Cumulants

TL;DR

This paper characterizes generating functions of functors on non-commutative probability spaces, establishing a unified framework for various types of non-commutative independence and their cumulants.

Contribution

It introduces a central lemma that links generating functions to universal products, enabling the construction of cumulants and Lie algebras for multiple non-commutative independences.

Findings

Existence of cumulants for various independences

Reconstruction of independence from cumulants and Lie algebras

Unified framework for different non-commutative independences

Abstract

In a central lemma we characterize "generating functions" of certain functors on the category of algebraic non-commutative probability spaces. Special families of such generating functions correspond to "unital, associative universal products" on this category, which again define a notion of non-commutative stochastic independence. Using the central lemma, we can prove the existence of cumulants and of "cumulant Lie-algebras" for a wide class of independences. These include the five independences (tensor, free, Boolean, monotone, anti-monotone) appearing in N. Murakis classification, c-free independence of M. Bozejko and R. Speicher, the indented product of T. Hasebe and the bi-free independence of D. Voiculescu. We show that the non-commutative independence can be reconstructed from its cumulants and cumulant Lie algebras.