Weight-preserving isomorphisms between spaces of continuous functions: The scalar case

TL;DR

This paper characterizes weight-preserving isomorphisms between spaces of continuous functions over locally compact spaces, extending Hamming metrics to infinite spaces and linking to coding theory.

Contribution

It extends the concept of Hamming metric to infinite spaces and characterizes isometries as weighted composition operators under mild conditions.

Findings

Hamming isometries can be represented as weighted composition operators

Extension of Hamming metric to infinite spaces

Connections established between isometries and coding theory

Abstract

Let $\mathbb F$ be a finite field and let $\mathcal A$ and $\mathcal B$ be vector spaces of $\mathbb F$-valued continuous functions defined on locally compact spaces $X$ and $Y$, respectively. We look at the representation of linear bijections $H:\mathcal A\longrightarrow \mathcal B$ by continuous functions $h:Y\longrightarrow X$ as weighted composition operators. In order to do it, we extend the notion of Hamming metric to infinite spaces. Our main result establishes that under some mild conditions, every Hamming isometry can be represented as a weighted composition operator. Connections to coding theory are also highlighted.