Accessible operators on ultraproducts of Banach spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917882916503552 |
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| author | Sánchez, Félix Cabello |
| author_facet | Sánchez, Félix Cabello |
| contents | We address a question by Henry Towsner about the possibility of representing linear operators between ultraproducts of Banach spaces by means of ultraproducts of nonlinear maps. We provide a bridge between these "accessible" operators and the theory of twisted sums through the so-called quasilinear maps. Thus, for many pairs of Banach spaces $X$ and $Y$, there is an "accessible" operator $X_U\to Y_U$ that is not the ultraproduct of a family of operators $X\to Y$ if and only if there is a short exact sequence of quasi-Banach spaces and operators $0\to Y\to Z\to X\to 0$ that does not split. We then adapt classical work by Ribe and Kalton--Peck to exhibit pretty concrete examples of accessible functionals and endomorphisms for the sequence spaces $\ell_p$. The paper is organized so that the main ideas are accessible to readers working on ultraproducts and requires only a rustic knowledge of Banach space theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01297 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Accessible operators on ultraproducts of Banach spaces Sánchez, Félix Cabello Functional Analysis Logic 46M07, 46M18, 46B08, 46A16 We address a question by Henry Towsner about the possibility of representing linear operators between ultraproducts of Banach spaces by means of ultraproducts of nonlinear maps. We provide a bridge between these "accessible" operators and the theory of twisted sums through the so-called quasilinear maps. Thus, for many pairs of Banach spaces $X$ and $Y$, there is an "accessible" operator $X_U\to Y_U$ that is not the ultraproduct of a family of operators $X\to Y$ if and only if there is a short exact sequence of quasi-Banach spaces and operators $0\to Y\to Z\to X\to 0$ that does not split. We then adapt classical work by Ribe and Kalton--Peck to exhibit pretty concrete examples of accessible functionals and endomorphisms for the sequence spaces $\ell_p$. The paper is organized so that the main ideas are accessible to readers working on ultraproducts and requires only a rustic knowledge of Banach space theory. |
| title | Accessible operators on ultraproducts of Banach spaces |
| topic | Functional Analysis Logic 46M07, 46M18, 46B08, 46A16 |
| url | https://arxiv.org/abs/2501.01297 |