ℵ-injective Banach spaces and ℵ-projective compacta

• Antonio Avilés ^[1] ; Félix Cabello Sánchez ^[2] ; Jesús M.F. Castillo ^[2] ; Manuel González ^[3] ; Yolanda Moreno ^[2]
  1. [1] Universidad de Murcia

     Universidad de Murcia

     Murcia, España

     
  2. [2] Universidad de Extremadura

     Universidad de Extremadura

     Badajoz, España

     
  3. [3] Universidad de Cantabria

     Universidad de Cantabria

     Santander, España

     

  Mostrar afiliaciones +
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 31, Nº 2, 2015, págs. 575-600
• Idioma: inglés
• Texto completo no disponible (Saber más ...)
• Resumen

  • 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 Fℵ-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 Fℵ-spaces.