This is an announcement for the paper "$\aleph$-injective Banach spaces and $\aleph$-projective compacta" by Antonio Aviles, Felix Cabello Sanchez, Jesus M. F. Castillo, Manuel Gonzalez and Yolanda Moreno.
Abstract: A Banach space $E$ is said to be injective if for every Banach space $X$ and every subspace $Y$ of $X$ every operator $t:Y\to E$ has an extension $T:X\to E$. We say that $E$ is $\aleph$-injective (respectively, universally $\aleph$-injective) if the preceding condition holds for Banach spaces $X$ (respectively $Y$) 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 $\aleph=\aleph_1$ is the first uncountable cardinal. When dealing with the corresponding ``isometric'' properties we arrive to our main examples: ultraproducts and spaces of type $C(K)$. We prove that ultraproducts built on countably incomplete $\aleph$-good ultrafilters are $(1,\aleph)$-injective as long as they are Lindenstrauss spaces. We characterize $(1,\aleph)$-injective $C(K)$ spaces as those in which the compact $K$ is an $F_\aleph$-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 $F_\aleph$-spaces.
Archive classification: math.FA
Mathematics Subject Classification: 46B03, 54B30, 46B08, 54C15, 46B26
Remarks: This paper is to appear in Revista Matem'atica Iberoamericana
Submitted from: castillo@unex.es
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1406.6733
or