Conditionally bi-free independence for pairs of faces

Abstract

In this paper, the notion of conditionally bi-free independence for pairs of algebras is introduced. The notion of conditional (ℓ, r)-cumulants are introduced and it is demonstrated that conditionally bi-free independence is equivalent to mixed cumulants. Furthermore, limit theorems for the additive conditionally bifree convolution are studied using both combinatorial and analytic techniques. In particular, a conditionally bi-free partial R-transform is constructed and a conditionally bi-free analogue of the Lévy-Hinčin formula for planar Borel probability measures is derived.

Bi-monotonic Independence for Pairs of Algebras

“…The notions of conditionally bi-free (c-bi-free for short) and bi-Boolean independences were introduced in [9,10] as generalizations of c-free and Boolean independences and are universal. For c-bi-free, given a pair of unital linear functionals ϕ k , ψ k : A k,ℓ * A k,r → C for each k ∈ K there is a unique pair of unital linear functionals ϕ, ψ : * k∈K (A k,ℓ * A k,r ) → C such that ϕ| A k,ℓ * A k,r = ϕ k , ψ| A k,ℓ * A k,r = ψ k , and {(A k,ℓ , A k,r )} k∈K is c-bi-freely independent with respect to (ϕ, ψ).…”

“…Recall as shown in [9,Corollary 5.7] that for a pair (a, b) in a double non-commutative space (A, ϕ, ψ), the reduced c-bi-free partial R-transform of (a, b) is given by…”

Free-Boolean independence for pairs of algebras

“…More recently, more independence relations for pairs of algebras are studied. For example in [3,4,5], conditionally bi-free independence, bi-Boolean independence, bi-monotone independence are introduced and studied.…”

Free-Boolean independence with amalgamation

“…Their operator-valued generalization were studied as well [9,12,13]. Recently, their corresponding independence relations for pairs of random variables, analog of Voiculescu's bi-free theory, were introduced and studied [4,3,6]. Furthermore, the conditionally bi-free independence with amalgamation is studied in [5].…”