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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.19642 |
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| _version_ | 1866908791559159808 |
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| author | Miura, Takeshi Shibata, Natsumi |
| author_facet | Miura, Takeshi Shibata, Natsumi |
| contents | 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 $τ\colon Y \to X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_19642 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Additive and multiplicative maps in norm on the positive cone of continuous function algebras Miura, Takeshi Shibata, Natsumi Functional Analysis 46E25, 46B04, 46J10, 47B33 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 $τ\colon Y \to X$. |
| title | Additive and multiplicative maps in norm on the positive cone of continuous function algebras |
| topic | Functional Analysis 46E25, 46B04, 46J10, 47B33 |
| url | https://arxiv.org/abs/2601.19642 |