Accessible operators on ultraproducts of Banach spaces

• Autores: Félix Cabello Sánchez
• Localización: Extracta mathematicae, ISSN-e 0213-8743, Vol. 39, Nº 2, 2024, págs. 207-234
• Idioma: inglés
• Enlaces
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• Resumen

  • 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 XU → YU that is not the ultraproduct of a family of operators X → Y if and only if there is a short exact sequence of quasi-Banach spaces and operators 0 → Y → Z → X → 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 lp 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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