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

TL;DR

This paper addresses how the norm structure on the positive cone $C_0^+(X)$ determines the topology of $X$ in locally compact Hausdorff spaces. It proves that any bijection $T:C_0^+(X)\to C_0^+(Y)$ preserving the norm of sums and products is a composition operator $T(f)=f\circ\tau$ associated with a homeomorphism $\tau:Y\to X$. The proof extends $T$ to a linear isometry on $C_0(X)$, uses the Banach-Stone theorem to deduce a representation with a sign map, and then shows the sign must be identically 1 due to positivity, hence $T(f)=f\circ\tau$ for all $f\in C_0^+(X)$. The result extends GK-orig-type theorems to positive cones and clarifies how norm-algebraic data encodes the underlying topology.

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 $T(f) = f \circ τ$, induced by a homeomorphism $τ\colon Y \to X$.

Theorems & Definitions (15)

• Theorem
• Lemma 1: Hirota Hirota
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof