Norm additive mappings between the positive cones of continuous function algebras

TL;DR

This paper characterizes bijections between positive cones of continuous functions vanishing at infinity that preserve a specific norm additive structure, revealing they are essentially weighted composition operators induced by homeomorphisms.

Contribution

It provides a complete description of norm additive bijections on positive cones of $C_0(X)$ spaces, extending analysis beyond the unital case to non-unital settings.

Findings

Such bijections are represented as weighted composition operators.

They are characterized by a homeomorphism and a positive continuous weight function.

The results apply to the non-unital $C_0(X)$ setting, unlike previous compact case analyses.

Abstract

We study bijections between the positive cones of spaces of continuous functions vanishing at infinity that satisfy a norm additive condition. Such maps arise naturally in the study of nonlinear functional equations and norm-preserving structures on function spaces. While in the compact (unital) case these maps can often be analyzed via linear extension techniques, the non-unital setting $C_0(X)$ requires a different approach due to the absence of a distinguished unit element. In this paper, we show that every bijection $T:C_0^+(X)\to C_0^+(Y)$ between the positive cones of $C_0(X)$ and $C_0(Y)$ satisfying \[ \|T(f+g)\|=\|Tf+Tg\| \] for all $f,g\in C_0^+(X)$ admits a representation of the form \[ Tf(y)=h(y)f(\tau(y)), \] where $\tau:Y\to X$ is a homeomorphism and $h$ is a bounded continuous function from $Y$ to $(0,\infty)$. This yields a complete characterization of norm additive bijections on positive cones of $C_0^+(X)$.