$\aleph$-injective Banach spaces and $\aleph$-projective compacta

TL;DR

This paper explores the properties of $\u2206$-injective Banach spaces and $\u2206$-projective compacta, extending classical concepts to uncountable cardinals, and characterizes these spaces using ultraproducts and $C(K)$ spaces.

Contribution

It extends the theory of injective Banach spaces to uncountable densities and characterizes $\u2206$-injective $C(K)$ spaces via $F_$-spaces, introducing new examples and properties.

Findings

Ultraproducts on countably incomplete ultrafilters are $(1,)$-injective if they are Lindenstrauss spaces.

Characterization of $(1,)$-injective $C(K)$ spaces as those with $K$ being an $F_06$-space.

Uncovered projectiveness properties of $F_06$-spaces.

Abstract

A Banach space $E$ is said to be injective if for every Banach space $X$ and every subspace $Y$ of $X$ every operator $t:Y\to E$ has an extension $T:X\to E$. We say that $E$ is $\aleph$-injective (respectively, universally $\aleph$-injective) if the preceding condition holds for Banach spaces $X$ (respectively $Y$) with density less than a given uncountable cardinal $\aleph$. We perform a study of $\aleph$-injective and universally $\aleph$-injective Banach spaces which extends the basic case where $\aleph=\aleph_1$ is the first uncountable cardinal. When dealing with the corresponding "isometric" properties we arrive to our main examples: ultraproducts and spaces of type $C(K)$. We prove that ultraproducts built on countably incomplete $\aleph$-good ultrafilters are $(1,\aleph)$-injective as long as they are Lindenstrauss spaces. We characterize $(1,\aleph)$-injective $C(K)$ spaces as those in which the compact $K$ is an $F_\aleph$-space (disjoint open subsets which are the union of less than $\aleph$ many closed sets have disjoint closures) and we uncover some projectiveness properties of $F_\aleph$-spaces.