Abstract. In this paper we deal with two weaker forms of injectivity which turn out to have a rich structure behind: separable injectivity and universal separable injectivity. We show several structural and stability properties of these classes of Banach spaces. We provide natural examples of (universally) separably injective spaces, including L∞ ultraproducts built over countably incomplete ultrafilters, in spite of the fact that these ultraproducts are never injective. We obtain two fundamental characterizations of universally separably injective spaces: a) A Banach space E is universally separably injective if and only if every separable subspace is contained in a copy of ℓ∞ inside E. b) A Banach space E is universally separably injective if and only if for every separable space S one has Ext(ℓ∞/S, E) = 0. The final Section of the paper focuses on special properties of 1-separably injective spaces. Lindenstrauss obtained in the middle sixties a result that can be understood as a proof that, under the continuum hypothesis, 1-separably injective spaces are 1-universally separably injective; he left open the question in ZFC. We construct a consistent example of a Banach space of type C(K) which is 1-separably injective but not 1-universally separably injective.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.