(\aleph\)-injective Banach spaces and \(\aleph\)-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 \(t:Y\to E\) has an extension \(T:X\to E\). We say that \(E\) is \(\aleph\)-injective (respectively, universally \(\aleph\)-injective) if the preceding condition holds for Ba...

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Bibliographic Details
Published in:arXiv.org 2014-06
Main Authors: Avilés, Antonio, Félix Cabello Sánchez, Castillo, Jesús M F, González, Manuel, Moreno, Yolanda
Format: Article
Language:English
Subjects:
Banach spaces
Closures
Error analysis
Online Access:Get full text
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Summary: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.
ISSN:2331-8422