How Free-Free-Boolean Independence Arises in Bi-Free Probability
Daniel Pepper
TL;DR
The work addresses how free-free-Boolean independence for triples of algebras (with amalgamation over a base algebra $B$) can be realized within bi-free probability, and how its cumulants arise from bi-free cumulants. It builds a comprehensive $B$-valued probabilistic framework, develops a robust LR-diagram calculus centered on bi-non-crossing partitions, and defines Boolean projections to model multi-face interactions. The main contribution is constructing abstract bi-free ffb $B$-systems and a detailed embedding that preserves joint distributions, enabling the free-free-Boolean cumulants to be derived from bi-free cumulants. This unifies multi-algebra independences under the bi-free umbrella and provides concrete combinatorial tools for calculating and understanding such independences with amalgamation over $B$.
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.