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arXiv · 2501.01297

Accessible operators on ultraproducts of Banach spaces

Félix Cabello Sánchez
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Abstract

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.

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Félix Cabello Sánchez. 2025-01-02. Accessible operators on ultraproducts of Banach spaces. https://arxiv.org/abs/2501.01297

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