Ring isomorphisms in norm between Banach algebras of continuous complex-valued functions
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914258764169216 |
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| author | Miura, T. Takahashi, T. |
| author_facet | Miura, T. Takahashi, T. |
| contents | Let $X$ and $Y$ be compact Hausdorff spaces, and let $C(X)$ and $C(Y)$ denote the commutative Banach algebras of all continuous complex-valued functions on $X$ and $Y$, respectively. We study bijective maps $T$ from $C(X)$ onto $C(Y)$ which preserve the ring structure in the norm in the following sense: \[ \|T(f+g)\|=\|T(f)+T(g)\|,\quad \|T(fg)\|=\|T(f)T(g)\| \qquad(f,g\in C(X)). \] Our main objective is to clarify whether such maps must necessarily be induced by homeomorphisms between the underlying spaces. Under the additional assumption that $T(\overline{f})=\overline{T(f)}$ for $f\in C(X)$, we prove that $T$ is a real-linear isometry. As a consequence, we obtain a concrete representation of such maps as weighted composition operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_11165 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Ring isomorphisms in norm between Banach algebras of continuous complex-valued functions Miura, T. Takahashi, T. Functional Analysis Let $X$ and $Y$ be compact Hausdorff spaces, and let $C(X)$ and $C(Y)$ denote the commutative Banach algebras of all continuous complex-valued functions on $X$ and $Y$, respectively. We study bijective maps $T$ from $C(X)$ onto $C(Y)$ which preserve the ring structure in the norm in the following sense: \[ \|T(f+g)\|=\|T(f)+T(g)\|,\quad \|T(fg)\|=\|T(f)T(g)\| \qquad(f,g\in C(X)). \] Our main objective is to clarify whether such maps must necessarily be induced by homeomorphisms between the underlying spaces. Under the additional assumption that $T(\overline{f})=\overline{T(f)}$ for $f\in C(X)$, we prove that $T$ is a real-linear isometry. As a consequence, we obtain a concrete representation of such maps as weighted composition operators. |
| title | Ring isomorphisms in norm between Banach algebras of continuous complex-valued functions |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2601.11165 |