Ferenczi, ValentinLópez Abad, Jorge2025-02-112025-02-112024-07-12Ferenczi, & Lopez-Abad. (2024). Envelopes in Banach spaces. Banach Journal of Mathematical Analysis, 18(3). https://doi.org/10.1007/S43037-024-00346-W2662-2033 | eISSN 1735-8787https://doi.org/10.1007/s43037-024-00346-whttps://hdl.handle.net/20.500.14468/25897The registered version of this article, first published in “Banach Journal of Mathematical Analysis, 18(3), paper No. 59", is available online at the publisher's website: Springer, https://orcid.org/0000-0001-5239-111XLa versión registrada de este artículo, publicado por primera vez en “Banach Journal of Mathematical Analysis, 18(3), paper No. 59", está disponible en línea en el sitio web del editor: Springer, https://orcid.org/0000-0001-5239-111XWe introduce the notion of isometric envelope of a subspace in a Banach space, establishing its connections with several key elements: (a) we explore its relation to the mean ergodic projection on fixed points within a semigroup of contractions, (b) we draw parallels with Korovkin sets from the 1970s, (c) we investigate its impact on the extension properties of linear isometric embeddings. We use this concept to address the recent conjecture that the Gurarij space and the spaces Lp, p∉2N+4 are the only separable approximately ultrahomogeneous Banach spaces (a certain multidimensional transitivity of the action of the linear isometry group). The similar conjecture for Fraïssé Banach spaces (a strengthening of the approximately homogeneous property) is also considered. We characterize the Hilbert space as the only separable reflexive space in which any closed subspace coincides with its envelope; and we show that the Gurarij space satisfies the same property. We compute some envelopes in the case of Lebesgue spaces, showing that the reflexive Lp-spaces are the only reflexive rearrangement invariant spaces on [0, 1] for which all 1-complemented subspaces are envelopes. We also identify the isometrically unique “full” quotient space of Lp by a Hilbertian subspace, for appropriate values of p, as well as the associated topological group embedding of the unitary group into the isometry group of Lp.esinfo:eu-repo/semantics/openAccess12 Matemáticasartículo