科研之友
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摘要
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.
- 出版日期2015-2
