$\aleph$-injective Banach spaces and $\aleph$-projective compacta

Abstract

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 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 = ℵ 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 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.

The separable Jung constant in Banach spaces

“…Indeed, J(c 0 ) = 2 = J s (c 0 ) while J( ∞ ) = J s ( ∞ ) = 1. In general, given a countably incomplete ultrafilter U on N, one has J s (C[0, 1]) = 2 but J s (C[0, 1] U ) = 1 since according to [6,7] the ultrapower of a Lindenstrauss space is 1-separably injective, as already mentioned. On the other hand, J( ∞ ) = 1 but J(( ∞ ) U ) = 2 since, again according to [6,7] no infinite-dimensional ultrapower is injective.…”

“…Two important variations of this notion are: λ-separable injectivity, when the property above holds when F is separable; and universal λ-separable injectivity [3,7], when the preceding property holds when E is separable. Other cardinal variations of this notion were studied in [6] and [7]. Accordingly, given an uncountable cardinal ℵ, a Banach space X is said to be (λ, ℵ)-injective if the preceding condition holds for Banach spaces F with density character less than ℵ.…”

On the uniqueness of cellular injectives

“…Then M-injective Banach spaces are precisely Banach spaces C(X ) where X is an extremally disconnected compact Hausdorff spaces and Ban has enough M-injectives (see [12]). But M is not cofibrantly generated because, for a set X of regular monomorphisms, X -injectives contain all (1, κ)injective Banach spaces where the codomains of morphisms of X are κ-presentable (see [6]).…”

Separably injective C*-algebras

“…Recently, separably injective Banach spaces have been studied in depth by Avilés, Cabello Sánchez, Castillo, González and Moreno in [2,3,4], where one can find a number of interesting examples of these spaces despite the scarcity of examples of injective Banach spaces. In contrast to the fact that 1-injective Banach spaces are isometric to the Banach space C(Ω) of continuous functions on a compact Hausdorff space Ω [12,14], 1-separably injective Banach spaces need not be complemented in C(Ω).…”