Invariant structure preserving functions and an Oka-Weil Kaplansky density type theorem

TL;DR

This paper develops a theory of invariant structure preserving functions on structured spaces, showing they can be approximated by polynomials and establishing a density theorem for free functions on polynomially convex operator domains.

Contribution

It introduces a new framework for invariant structure preserving functions and proves a density theorem extending classical approximation results to free functions.

Findings

Invariant structure preserving functions are pointwise approximable by polynomials.

On polynomially convex operator domains, contractive free functions can be approximated by contractive polynomials.

The paper extends classical approximation theorems to the setting of free functions and structured topological spaces.

Abstract

We develop the theory of invariant structure preserving and free functions on a general structured topological space. We show that an invariant structure preserving function is pointwise approximiable by the appropriate analog of polynomials in the strong topology and therefore a free function. Moreover, if a domain of operators on a Hilbert space is polynomially convex, the set of free functions satisfies a Oka-Weil Kaplansky density type theorem -- contractive functions can be approximated by contractive polynomials.