Banach spaces of universal disposition

Abstract

In this paper we present a method to obtain Banach spaces of universal and almost-universal disposition with respect to a given class $\mathfrak{M}$ of normed spaces. The method produces, among other, the Gurari\uı space $\mathcal{G}$ (the only separable Banach space of almost-universal disposition with respect to the class $\mathfrak{F}$ of finite dimensional spaces), or the Kubis space $\mathcal{K}$ (under {\sf CH}, the only Banach space with the density character the continuum which is of universal disposition with respect to the class $\mathfrak{S}$ of separable spaces). We moreover show that $\mathcal{K}$ is not isomorphic to a subspace of any $C(K)$-space -- which provides a partial answer to the injective space problem-- and that --under {\sf CH}-- it is isomorphic to an ultrapower of the Gurari\uı space. We study further properties of spaces of universal disposition: separable injectivity, partially automorphic character and uniqueness properties.

References in corpus (1)

• On separably injective Banach spaces and Corrigendum to "On separably injective Banach spaces" [Adv. Math. 234 (2013) 192--216]

Cited by in corpus (15)

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• Quasi-Banach spaces of almost universal disposition
• Metric-enriched categories and approximate Fraïssé limits
• Injective objects and retracts of Fraïssé limits
• A universal operator on the Gurarii space
• Game-theoretic characterization of the Gurarii space
• A separable Fréchet space of almost universal disposition
• Categories with norms
• A 1-separably injective space that does not contain ${\ell }_{\infty }$
• On non-archimedean Gurarii spaces
• 1-complemented subspaces of Banach spaces of universal disposition
• Isometric uniqueness of a complementably universal Banach space for Schauder decompositions
• A Separable Universal Homogeneous Banach Lattice
• On ultrapowers of Banach spaces of type ${\mathcal{L}}_{\infty }$
• A note on commutative and noncommutative Gurariy spaces