A Banach space is said to be injective if for every Banach space and every subspace of every operator has an extension . We say that is -injective (respectively, universally -injective) if the preceding condition holds for Banach spaces (respectively ) with density less than a given uncountable cardinal . We perform a study of -injective and universally -injective Banach spaces which extends the basic case where is the first uncountable cardinal. When dealing with the corresponding "isometric" properties we arrive to our main examples: ultraproducts and spaces of type . We prove that ultraproducts built on countably incomplete -good ultrafilters are -injective as long as they are Lindenstrauss spaces. We characterize -injective spaces as those in which the compact is an -space (disjoint open subsets which are the union of less than many closed sets have disjoint closures) and we uncover some projectiveness properties of -spaces.
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Antonio Avilés, Félix Cabello Sánchez, Jesús M.F. Castillo, Manuel González, Yolanda Moreno, -injective Banach spaces and -projective compacta. Rev. Mat. Iberoam. 31 (2015), no. 2, pp. 575–600DOI 10.4171/RMI/845