Norm additive mappings between the positive cones of continuous function algebras
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917447175503872 |
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| author | Shibata, Natsumi Miura, Takeshi |
| author_facet | Shibata, Natsumi Miura, Takeshi |
| contents | 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 \[ Tf(y)=h(y)f(τ(y)), \] where $τ:Y\to X$ 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 $C_0^+(X)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_26540 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Norm additive mappings between the positive cones of continuous function algebras Shibata, Natsumi Miura, Takeshi Functional Analysis 47B48, 46J10, 47B33, 39B52 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 \[ Tf(y)=h(y)f(τ(y)), \] where $τ:Y\to X$ 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 $C_0^+(X)$. |
| title | Norm additive mappings between the positive cones of continuous function algebras |
| topic | Functional Analysis 47B48, 46J10, 47B33, 39B52 |
| url | https://arxiv.org/abs/2604.26540 |