A Note on the Free and Cyclic Differential Calculus

TL;DR

This paper extends Voiculescu's algebraic characterization of cyclic gradients to free gradients and generalizes these results within multivariable difference quotient rings, introducing divergence operators and a unified framework.

Contribution

It introduces divergence operators for free and cyclic gradients and generalizes the algebraic characterizations within Voiculescu's framework, broadening the scope of differential calculus in noncommutative settings.

Findings

Characterization of free gradients analogous to cyclic gradients.

Development of divergence operators for both gradient types.

Unified framework encompassing polynomial and stochastic integral examples.

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.