Integrability of Free Noncommutative Functions

TL;DR

This paper develops a noncommutative analog of the Frobenius integrability theorem, providing conditions for higher order free noncommutative functions to possess an antiderivative, advancing the mathematical theory of noncommutative functions.

Contribution

It establishes necessary and sufficient conditions for the integrability of higher order free noncommutative functions, extending classical results to the noncommutative setting.

Findings

Derived conditions for integrability of noncommutative functions

Extended Frobenius theorem to noncommutative functions

Enhanced understanding of noncommutative calculus

Abstract

Noncommutative functions are graded functions between sets of square matrices of all sizes over two vector spaces that respect direct sums and similarities. They possess very strong regularity properties (reminiscent of the regularity properties of usual analytic functions) and admit a good difference-differential calculus. Noncommutative functions appear naturally in a large variety of settings: noncommutative algebra, systems and control, spectral theory, and free probability. Starting with pioneering work of J.L. Taylor, the theory was further developed by D.-V. Voiculescu, and established itself in recent years as a new and extremely active research area. The goal of the present paper is to establish a noncommutative analog of the Frobenius integrability theorem: we give necessary and sufficient conditions for higher order free noncommutative functions to have an antiderivative.