Abstract: 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 N-injective (respectively, universally N-injective) if the preceding condition holds for Banach spaces X (respectively Y) with density less than a given uncountable cardinal N. We perform a study of N-injective and universally N-injective Banach spaces which extends the basic case where N = N-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 N-good ultrafilters are (1, N)-injective as long as they are Lindenstrauss spaces. We characterize (1, N)-injective C(K) spaces as those in which the compact K is an F-N-space (disjoint open subsets which are the union of less than N many closed sets have disjoint closures) and we uncover some projectiveness properties of F-N-spaces.

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