A Note on the Free and Cyclic Differential Calculus
Abstract
In 2000, Voiculescu proved an algebraic characterization of cyclic gradients of noncommutative polynomials. We extend this remarkable result in two different directions: first, we obtain an analogous characterization of free gradients; second, we lift both of these results to Voiculescu's fundamental framework of multivariable generalized difference quotient rings. For that purpose, we develop the concept of divergence operators, for both free and cyclic gradients, and study the associated (weak) grading and cyclic symmetrization operators, respectively. One the one hand, this puts a new complexion on the initial polynomial case, and on the other hand, it provides a uniform framework within which also other examples - such as a discrete version of the It\^o stochastic integral - can be treated.
Cite
@article{arxiv.1910.07570,
title = {A Note on the Free and Cyclic Differential Calculus},
author = {Tobias Mai and Roland Speicher},
journal= {arXiv preprint arXiv:1910.07570},
year = {2020}
}
Comments
We have undertaken a minor revision; the paper has been accepted for publication in the special issue of the Journal of Operator Theory on the occasion of the 70th anniversary of Dan Voiculescu