ℵ-injective Banach spaces and ℵ-projective compacta
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Antonio Avilés [1] ; Félix Cabello Sánchez [2]
;
Jesús M.F. Castillo
[2]
;
Manuel González
[3]
;
Yolanda Moreno
[2]
-
[1] Universidad de Murcia
Universidad de Murcia
Murcia, España
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[2] Universidad de Extremadura
Universidad de Extremadura
Badajoz, España
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[3] Universidad de Cantabria
Universidad de Cantabria
Santander, España
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- Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 31, Nº 2, 2015, págs. 575-600
- Idioma: inglés
- DOI: 10.4171/RMI/845
- Texto completo no disponible (Saber más ...)
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Resumen
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=ℵ1 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 Fℵ-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 Fℵ-spaces.
