- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:48
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
31
2015
2
$\aleph$-injective Banach spaces and $\aleph$-projective compacta
Antonio
Avilés
Universidad de Murcia, MURCIA, SPAIN
Félix
Cabello Sánchez
Universidad de Extremadura, BADAJOZ, SPAIN
Jesús
Castillo
Universidad de Extremadura, BADAJOZ, SPAIN
Manuel
González
Universidad de Cantabria, SANTANDER, SPAIN
Yolanda
Moreno
Universidad de Extremadura, CACERES, SPAIN
Injective Banach spaces, cardinality assumptions, projective compacta
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.
Functional analysis
General topology
575
600
10.4171/RMI/845
http://www.ems-ph.org/doi/10.4171/RMI/845