Bi-Boolean independence for pairs of algebras

TL;DR

This paper introduces bi-Boolean independence for pairs of algebras, develops associated cumulants and transforms, and establishes limit theorems and formulas analogous to free probability, extending Boolean independence concepts.

Contribution

It extends Boolean independence to pairs of algebras, defines bi-Boolean cumulants and transforms, and derives a Lévy-Hinčin formula for bi-Boolean convolution.

Findings

Bi-Boolean independence characterized by vanishing mixed cumulants.

Construction of a bi-Boolean partial η-transform for limit theorems.

Derivation of a bi-Boolean Lévy-Hinčin formula.

Abstract

In this paper, the notion of bi-Boolean independence for non-unital pairs of algebras is introduced thereby extending the notion of Boolean independence to pairs of algebras. The notion of B-$(\ell, r)$-cumulants is defined via a bi-Boolean moment-cumulant formula over the lattice of bi-interval partitions, and it is demonstrated that bi-Boolean independence is equivalent to the vanishing of mixed B-$(\ell, r)$-cumulants. Furthermore, some of the simplest bi-Boolean convolutions are considered, and a bi-Boolean partial $\eta$-transform is constructed for the study of limit theorems and infinite divisibility with respect to the additive bi-Boolean convolution. In particular, a bi-Boolean L\'{e}vy-Hin\v{c}in formula is derived in perfect analogy with the bi-free case, and some Bercovici-Pata type bijections are provided. Additional topics considered include the additive bi-Fermi convolution, some relations between the $(\ell, r)$- and B-$(\ell, r)$-cumulants, and bi-Boolean independence in an amalgamated setting. The last section of this paper also includes an errata that will be published with this copy of the paper.