Double-ended queues and joint moments of left-right canonical operators on full Fock space

Abstract

We follow the guiding line offered by canonical operators on the full Fock space, in order to identify what kind of cumulant functionals should be considered for the concept of bi-free independence introduced in the recent work of Voiculescu. By following this guiding line we arrive to consider, for a general noncommutative probability space (A, phi), a family of "(l,r)-cumulant functionals" which enlarges the family of free cumulant functionals of the space. In the motivating case of canonical operators on the full Fock space we find a simple formula for a relevant family of (l,r)-cumulants of a (2d)-tuple (A_1, ..., A_d, B_1, ..., B_d), with A_1, ... , A_d canonical operators on the left and B_1, ... , B_d canonical operators on the right. This extends a known one-sided formula for free cumulants of A_1, ..., A_d, which establishes a basic operator model for the R-transform of free probability.

In this (final) version, the introduction was re-written to better show the motivation for the question considered in the paper

References in corpus (1)

• Free Probability for Pairs of Faces I

Cited by in corpus (14)

• Combinatorics of Bi-Freeness with Amalgamation
• On two-faced families of non-commutative random variables
• Conditionally Bi-Free Independence for Pairs of Algebras
• On Operator-Valued Bi-Free Distributions
• Independences and Partial $R$-Transforms in Bi-Free Probability
• A Combinatorial Approach to Voiculescu's Bi-Free Partial Transforms
• An alternating moment condition for bi-freeness
• Bi-Boolean independence for pairs of algebras
• Bi-monotonic independence for pairs of algebras
• Conditionally Bi-Free Independence with Amalgamation
• An operator that relates to semi-meander polynomials via a two-sided q-Wick formula
• Compound Bi-free Poisson Distributions
• Analytic aspects of the bi-free partial R-transform
• Two-faced Families of Non-commutative Random Variables Having Bi-free Infinitely Divisible Distributions