Analysis seminar

Alexandru Nica, Department of Pure Mathematics, University of Waterloo

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

One of the most basic problems of free probability is how to calculate the moments of a sum of two free elements in a noncommutative probability space. This is solved by using the R-transforms of the elements in question. Among the possible approaches to the R-transform of an element x, the very first one that was found (Voiculescu, 1986) goes by replacing x with a ”canonical operator” X which has the same moments as x, and where X is written in terms of the shift operator on l2. This approach extends to the case of a d-tuple of elements x1, ..., xd – the given d-tuple is replaced by a d-tuple of canonical operators X1,...,Xd, with the Xi’s written in terms of left-creation operators on the full Fock space F over Cd.

In 2013, Voiculescu introduced the concept of bi-freeness for elements in a noncommutative probability space; this calls for studying (2d)-tuples of canonical operators X1, ..., Xd, Y1, ..., Yd, where X1, ..., Xd are acting on the left and Y1, ..., Yd are acting on the right, on the same full Fock space F as above. In this talk I will present a recent joint work done with Mitja Mastnak (arXiv:1312.0269), where we study such (2d)-tuples of canonical operators.