How Free-Free-Boolean Independence Arises in Bi-Free Probability

TL;DR

This paper explores how free-free-Boolean independence for triples of algebras can be embedded into bi-free probability, allowing for new insights and cumulant constructions within this framework.

Contribution

It introduces a method to embed free-free-Boolean independence into bi-free probability and constructs its cumulants from bi-free cumulants, advancing the understanding of multi-algebra independence.

Findings

Free-free-Boolean independence can be embedded into bi-free probability.

Cumulants of free-free-Boolean independence can be derived from bi-free cumulants.

Provides a new perspective on multi-algebra independence in non-commutative probability.

Abstract

This work concerns notions of multi-algebra independence introduced by Liu and how they can be studied in the context of bi-free probability. In particular, we show how the free-free-Boolean independence for triples of algebras can be embedded intro and therefore studied from a lens of bi-free probability. It is also shown how its cumulants can be constructed from the bi-free cumulants.