# $ℵ$-injective Banach spaces and $ℵ$-projective compacta

### Antonio Avilés

Universidad de Murcia, Spain### Félix Cabello Sánchez

Universidad de Extremadura, Badajoz, Spain### Jesús M.F. Castillo

Universidad de Extremadura, Badajoz, Spain### Manuel González

Universidad de Cantabria, Santander, Spain### Yolanda Moreno

Universidad de Extremadura, Caceres, Spain

## Abstract

A Banach space $E$ is said to be injective if for every Banach space $X$ and every subspace $Y$ of $X$ every operator $t:Y→E$ has an extension $T:X→E$. We say that $E$ is $ℵ$-injective (respectively, universally $ℵ$-injective) if the preceding condition holds for Banach spaces $X$ (respectively $Y$) with density less than a given uncountable cardinal $ℵ$. We perform a study of $ℵ$-injective and universally $ℵ$-injective Banach spaces which extends the basic case where $ℵ=ℵ_{1}$ 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 $ℵ$-good ultrafilters are $(1,ℵ)$-injective as long as they are Lindenstrauss spaces. We characterize $(1,ℵ)$-injective $C(K)$ spaces as those in which the compact $K$ is an $F_{ℵ}$-space (disjoint open subsets which are the union of less than $ℵ$ many closed sets have disjoint closures) and we uncover some projectiveness properties of $F_{ℵ}$-spaces.

## Cite this article

Antonio Avilés, Félix Cabello Sánchez, Jesús M.F. Castillo, Manuel González, Yolanda Moreno, $ℵ$-injective Banach spaces and $ℵ$-projective compacta. Rev. Mat. Iberoam. 31 (2015), no. 2, pp. 575–600

DOI 10.4171/RMI/845