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arXiv · 2601.19642

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

Takeshi Miura·Natsumi Shibata
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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 $\tau\colon Y \to X$.

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Takeshi Miura, Natsumi Shibata. 2026-01-27. Additive and multiplicative maps in norm on the positive cone of continuous function algebras. https://arxiv.org/abs/2601.19642

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