Additive and multiplicative maps in norm on the positive cone of continuous function algebras

TL;DR

This paper characterizes bijections on positive continuous functions that preserve additive and multiplicative norms, showing they are induced by homeomorphisms between the underlying spaces.

Contribution

It proves that such norm-preserving maps are precisely composition operators induced by homeomorphisms, providing a structural characterization.

Findings

The map T is a composition operator induced by a homeomorphism.

T preserves the norm of sums and products of functions.

Such maps are bijections on the positive cone of continuous functions.

Abstract

Let $X$ and $Y$ be locally compact Hausdorff spaces. We denote by $C_0^+(X)$ the positive cone of all real-valued continuous functions on $X$ vanishing at infinity. In this paper, we consider a bijection $T\colon C_0^+(X) \to C_0^+(Y)$ satisfying the following two norm conditions for all $f, g \in C_0^+(X)$: \[ \|T(f+g)\| = \|T(f)+T(g)\|,\qquad \|T(f \cdot g)\| = \|T(f) \cdot T(g)\|. \] The main result of this paper is that such a map $T$ is a composition operator of the form $T(f) = f \circ \tau$, induced by a homeomorphism $\tau\colon Y \to X$.