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

TL;DR

This paper explores weaker forms of injectivity in Banach spaces, namely separable and universal separable injectivity, providing structural characterizations, examples, and examining their properties and distinctions.

Contribution

It introduces and characterizes separable and universal separable injectivity, offering new examples and analyzing their properties, including the impact of set-theoretic assumptions.

Findings

Universal separable injectivity characterized by containing $oldsymbol{ ext{ extltilde}}$ in every separable subspace

Examples of ultraproducts that are separably injective but not injective

Under continuum hypothesis, universal separable injectivity is not a 3-space property

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 $\mathcal L_\infty$ 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.