2011/03/30 by Antonio Avilés, Aviles, Antonio, Félix Cabello +7
Mathematics · #46A22 #46B04 #46B08 #46B26 #Advanced Banach Space Theory #FOS: Mathematics #Functional Analysis (math.FA)
paper · pdf · doi:10.48550/arxiv.1103.6064
openalex publication_date 2011/03/30 · openalex created_date 2017/04/07 · openalex updated_date 2026/07/28
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 \mathcal 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 \sf 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. We show that, under the continuum hypothesis, "to be universally separably injective" is not a 3-space property, as we wrongly claimed in the paper mentioned in the title.