Small Banach bundles and modules
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916268945178624 |
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| author | Chirvasitu, Alexandru |
| author_facet | Chirvasitu, Alexandru |
| contents | We characterize those (continuously-normed) Banach bundles $\mathcal{E}\to X$ with compact Hausdorff base whose spaces $Γ(\mathcal{E})$ of global continuous sections are topologically finitely-generated over the function algebra $C(X)$, answering a question of I. Gogić's and extending analogous work for metrizable $X$. Conditions equivalent to topological finite generation include: (a) the requirement that $\mathcal{E}$ be locally trivial and of finite type along locally closed and relatively $F_σ$ strata in a finite stratification of $X$; (b) the decomposability of arbitrary elements in $\ell^p(Γ(\mathcal{E}))$, $1\le p<\infty$ as sums of $\le N$ products in $\ell^p(C(X))\cdot Γ(\mathcal{E})$ for some fixed $N$; (c) the analogous decomposability requirement for maximal Banach-module tensor products $F\widehat{\otimes}_{C(X)}Γ(\mathcal{E})$ or (d) equivalently, only for $F=\ell^1(C(X))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_14518 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Small Banach bundles and modules Chirvasitu, Alexandru Functional Analysis General Topology Operator Algebras 46H25, 46J10, 46H10, 46M20, 54C40, 54D20, 54E35 We characterize those (continuously-normed) Banach bundles $\mathcal{E}\to X$ with compact Hausdorff base whose spaces $Γ(\mathcal{E})$ of global continuous sections are topologically finitely-generated over the function algebra $C(X)$, answering a question of I. Gogić's and extending analogous work for metrizable $X$. Conditions equivalent to topological finite generation include: (a) the requirement that $\mathcal{E}$ be locally trivial and of finite type along locally closed and relatively $F_σ$ strata in a finite stratification of $X$; (b) the decomposability of arbitrary elements in $\ell^p(Γ(\mathcal{E}))$, $1\le p<\infty$ as sums of $\le N$ products in $\ell^p(C(X))\cdot Γ(\mathcal{E})$ for some fixed $N$; (c) the analogous decomposability requirement for maximal Banach-module tensor products $F\widehat{\otimes}_{C(X)}Γ(\mathcal{E})$ or (d) equivalently, only for $F=\ell^1(C(X))$. |
| title | Small Banach bundles and modules |
| topic | Functional Analysis General Topology Operator Algebras 46H25, 46J10, 46H10, 46M20, 54C40, 54D20, 54E35 |
| url | https://arxiv.org/abs/2405.14518 |