Differential calculus for fully matricial functions I

TL;DR

This paper introduces a cyclic derivative for fully matricial functions, explores its properties, proves a Poincaré lemma, and clarifies its relation to existing non-commutative function frameworks.

Contribution

It develops a cyclic derivative concept for fully matricial functions and connects Voiculescu's framework with other non-commutative function theories.

Findings

Established a cyclic derivative for fully matricial functions

Proved the Poincaré lemma for certain classes of these functions

Clarified the relationship between different frameworks of nc functions

Abstract

We will introduce a cyclic derivative for fully (stably) matricial functions and study its basic properties. In particular, we will show the Poincar\'{e} lemma for stably matricial functions of certain classes. We will also position Voiculescu's framework of fully matricial functions in the context of nc functions due to Kaliuzhnyi-Verbovetskyi and Vinnikov in order to clarify the relation between the present work and previous related works.