Operator NC functions

TL;DR

This paper develops a new theory of non-commutative (NC) functions on certain von Neumann algebras, establishing derivative properties, approximation by free polynomials, and realization formulas without relying on matricial case results.

Contribution

It introduces a direct approach to NC functions on von Neumann algebras, proving derivative structure and approximation results independently of matricial case assumptions.

Findings

The $k$-th directional derivative is a $k$-linear homogeneous polynomial.

NC functions can be uniformly approximated by free polynomials.

Realization formulas are established for bounded NC functions on specific sets.

Abstract

We establish a theory of NC functions on a class of von Neumann algebras with a particular direct sum property, e.g. $B(\mathcal{H})$. In contrast to the theory's origins, we do not rely on appealing to results from the matricial case. We prove that the $k^{\text{th}}$ directional derivative of any NC function at a scalar point is a $k$-linear homogeneous polynomial in its directions. Consequences include the fact that NC functions defined on domains containing scalar points can be uniformly approximated by free polynomials as well as realization formulas for NC functions bounded on particular sets, e.g. the non-commutative polydisk and non-commutative row ball.