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

• Published in: arXiv.org 2017-03
• Main Authors: Aviles, Antonio, Cabello, Felix, Castillo, Jesus M F, Gonzalez, Manuel, Moreno, Yolanda
• Format: Article
• Language: English
• Subjects: Banach spaces; Error analysis; Hypotheses; Structural stability
• ISSN: 2331-8422

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 \(\mathcal L_\infty\) 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 \(\ell_\infty\) inside \(E\). b) A Banach space \(E\) is universally separably injective if and only if for every separable space \(S\) one has \(\Ext(\ell_\infty/S, E)=0\). 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.