Multi-Algebra Independence and Bi-Free Probability

Abstract

Non-commutative probability is an abstraction of classical probability that arises from considering random variables that do not commute in multiplication. In this non-commutative setting, five different types of independence between variables arise that are unique but closely parallel the classical independence. For example, each type of independence has its own theory of probability associated including central limit theorems and stochastic processes. More recently an independence between pairs of variables known as bi-free independence has received some attention. It is a remarkable fact that each of the five main types of independence can be embedded into this bi-free independence. Shortly after the inception of bi-free independence, a generalization was introduced, spawning simultaneously several notions of independence between tuples of variables. This wave of new types of independence appears to have dramatically increased the potential avenues of research. One is left with the question of how these can be corralled, if possible. Given the embeddability of the five main types of independence into bi-free probability, one naturally extends the question of embeddability to the more recent multi-algebra independences. Can these too be studied from the context of bi-free probability? The main results of this work will be an affirmative answer for a small collection of these, most notably the free-Boolean and free-free-Boolean independences for pairs and triples of algebras respectively. Some other simple embeddings are shown, as well as some partial negative results.