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On Operator-Valued Bi-Free Distributions

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 $(B, B^{\mathrm{op}})$ 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.