On separably injective Banach spaces

We deal with two weak 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∞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 EE is universally separably injective if and only if every separable subspace is contained in a copy of ℓ∞ℓ∞ inside EE. (b) A Banach space EE is universally separably injective if and only if for every separable space SS one has Ext(ℓ∞/S,E)=0Ext(ℓ∞/S,E)=0. Section 6 focuses on special properties of 1-separably injective spaces. Lindenstrauss proved in the middle sixties that, under CH, 1-separably injective spaces are 1-universally separably injective and left open the question in ZFC. We construct a consistent example of a Banach space of type C(K)C(K) which is 1-separably injective but not universally 1-separably injective.