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, φ), 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 (A1,…,Ad, B1,…,Bd), with A1,…,Ad canonical operators on the left and B1,…,Bd canonical operators on the right. This extends a known one-sided formula for free cumulants of A1,…,Ad, which establishes a basic operator model for the R-transform of free probability.