# Revista Matemática Iberoamericana

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**Volume 31, Issue 2, 2015, pp. 575–600**

**DOI: 10.4171/RMI/845**

Published online: 2015-07-16

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

Antonio Avilés^{[1]}, Félix Cabello Sánchez

^{[2]}, Jesús M.F. Castillo

^{[3]}, Manuel González

^{[4]}and Yolanda Moreno

^{[5]}(1) Universidad de Murcia, Spain

(2) Universidad de Extremadura, Badajoz, Spain

(3) Universidad de Extremadura, Badajoz, Spain

(4) Universidad de Cantabria, Santander, Spain

(5) Universidad de Extremadura, Caceres, Spain

A Banach space $E$ is said to be injective if for every Banach space $X$ and every subspace $Y$ of $X$ every operator $t\colon Y\to E$ has an extension $T\colon X \to E$. 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.

*Keywords: *Injective Banach spaces, cardinality assumptions, projective compacta

Avilés Antonio, Cabello Sánchez Félix, Castillo Jesús, González Manuel, Moreno Yolanda: $\aleph$-injective Banach spaces and $\aleph$-projective compacta. *Rev. Mat. Iberoam.* 31 (2015), 575-600. doi: 10.4171/RMI/845