JournalsrmiVol. 31, No. 2pp. 575–600

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

  • Antonio Avilés

    Universidad de Murcia, Spain
  • Félix Cabello Sánchez

    Universidad de Extremadura, Badajoz, Spain
  • Jesús M.F. Castillo

    Universidad de Extremadura, Badajoz, Spain
  • Manuel González

    Universidad de Cantabria, Santander, Spain
  • Yolanda Moreno

    Universidad de Extremadura, Caceres, Spain
$\aleph$-injective Banach spaces and $\aleph$-projective compacta cover
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Abstract

A Banach space EE is said to be injective if for every Banach space XX and every subspace YY of XX every operator t ⁣:YEt\colon Y\to E has an extension T ⁣:XET\colon X \to E. We say that EE is \aleph-injective (respectively, universally \aleph-injective) if the preceding condition holds for Banach spaces XX (respectively YY) with density less than a given uncountable cardinal \aleph. We perform a study of \aleph-injective and universally \aleph-injective Banach spaces which extends the basic case where =1\aleph=\aleph_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)C(K). We prove that ultraproducts built on countably incomplete \aleph-good ultrafilters are (1,)(1,\aleph)-injective as long as they are Lindenstrauss spaces. We characterize (1,)(1,\aleph)-injective C(K)C(K) spaces as those in which the compact KK is an FF_\aleph-space (disjoint open subsets which are the union of less than \aleph many closed sets have disjoint closures) and we uncover some projectiveness properties of FF_\aleph-spaces.

Cite this article

Antonio Avilés, Félix Cabello Sánchez, Jesús M.F. Castillo, Manuel González, Yolanda Moreno, \aleph-injective Banach spaces and \aleph-projective compacta. Rev. Mat. Iberoam. 31 (2015), no. 2, pp. 575–600

DOI 10.4171/RMI/845