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Double-ended queues and joint moments of left-right canonical operators on full Fock space
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.Comment: In this (final) version, the introduction was re-written to better
show the motivation for the question considered in the pape
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