In this paper, we develop the theory of bi-freeness in an amalgamated setting. We construct the operator-valued bi-free cumulant functions, and show that the vanishing of mixed cumulants is necessary and sufficient for bi-free independence. Further, we develop a multiplicative convolution for operator-valued random variables and explore ways to construct bi-free pairs of B-faces.Comment: 35 page

Abstract. We demonstrate that the notions of bi-free independence and combinatorialbi-free independence of two-faced families are equivalent using a diagrammatic view of bi-non-crossing partitions. These diagrams produce an operator model on a Fock space suitable for representing any two-faced family of non-commutative random variables. Furthermore, using a Kreweras complement on bi-non-crossing partitions we establish the expected formulas for the multiplicative convolution of a bi-free pair of two-faced families.

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.

In this paper, operator-valued bi-free distributions are investigated. Given a subalgebra D of a unital algebra B, it is established that a two-faced family Z is bi-free from (B, B op ) over D if and only if certain conditions relating the B-valued and D-valued bi-free cumulants of Z are satisfied. Using this, we verify that a two-faced family of matrices is R-cyclic if and only if they are bi-free from the scalar matrices over the scalar diagonal matrices. Furthermore, the operator-valued bi-free partial R-, S-, and T -transforms are constructed. New proofs of results from free probability are developed in order to facilitate many of these bi-free results.for all T ∈ L(M); that is, apply T to 1 M and take the expectation of M onto N. Further define *homomorphisms L : M → L(M) and R : M op → L(M) by L(X)(A) = XA and R(X)(A) = AX for all X, A ∈ M. For this discussion we will call L(X) a left operator and R(X) a right operator. If M = M 1 * N M 2 , the amalgamated free product of von Neumann algebras M 1 and M 2 over N, then the Date: July 9, 2018. 2010 Mathematics Subject Classification. 46L54, 46L53. Key words and phrases. bi-free probability, operator-valued distributions, amalgamating over a subalgebra, R-cyclic matrices, R-transform, S-transform. PAUL SKOUFRANIS Given χ : {1, . . . , n} → {ℓ, r} withdefine the permutation s χ on {1, . . . , n} by s χ (q) = k q . The only differences between the combinatorial aspects of free and bi-free probability arise from dealing with s χ . Using s χ , define the total ordering ≺ χ on {1, . . . , n} by k 1 ≺ χ k 2 if and only if s −1 χ (k 1 ) < s −1 χ (k 2 ). Instead of reading {1, . . . , n} in the traditional order, ≺ χ corresponds to reading χ −1 ({ℓ}) in increasing order followed by reading χ −1 ({r}) in decreasing order.A subset V ⊆ {1, . . . , n} is said to be a χ-interval if V is an interval with respect to the ordering ≺ χ . In addition, min ≺χ (V ) and max ≺χ (V ) denote the minimal and maximal elements of V with respect to the ordering ≺ χ .Definition 2.1. A partition π ∈ P(n) is said to be bi-non-crossing with respect to χ if the partition s −1 χ · π (the partition formed by applying s −1 χ to each entry of each block of π) is non-crossing. Equivalently π is bi-non-crossing if whenever there are blocks U, V ∈ π with u 1

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