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 ([alpha], [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[subscript 1], . . . , A[subscript d], B[subscript 1], . . . , B[subscript d]), with A[subscript 1], . . . , A[subscript d] canonical operators on the left and B[subscript 1], . . . , B[subscript d] canonical operators on the right. This extends a known one-sided formula for free cumulants of A[subscript 1], . . . , A[subscript d], which establishes a basic operator model for the R-transform of free probability.