Non-commutative stochastic independence and cumulants

Abstract

In a fundamental 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 fundamental lemma, we prove the existence of cumulants and of “cumulant Lie algebras” for all independences coming from a unital, associative universal product. These include the five independences (tensor, free, Boolean, monotone, anti-monotone) appearing in Muraki’s classification, c-free independence of Bożejko and Speicher, the indented product of Hasebe and the bi-free independence of Voiculescu. We show how the non-commutative independence can be reconstructed from its cumulants and cumulant Lie algebras.