2014/06/25 by Antonio Avilés, Avilés, Antonio, Félix Cabello Sánchez +7 · 1 citation
Mathematics · #46B03 #46B08 #46B26 #54B30 #54C15 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:46B03 #msc:46B08 #msc:46B26 #msc:54B30 #msc:54C15
paper · pdf · doi:10.48550/arxiv.1406.6733
This paper is to appear in Revista Matemática Iberoamericana
arxiv created 2014/06/25 · arxiv updated 2014/06/27
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 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.