This is an announcement for the paper "On separably injective Banach spaces" by Antonio Aviles, Felix Cabello, Jesus M. F. Castillo, Manuel Gonzalez, and Yolanda Moreno.

Abstract: In this paper we deal with two weaker forms of injectivity which turn out to have a rich structure behind: separable injectivity and universal separable injectivity. We show several structural and stability properties of these classes of Banach spaces. We provide natural examples of (universally) separably injective spaces, including $\mathcal L_\infty$ ultraproducts built over countably incomplete ultrafilters, in spite of the fact that these ultraproducts are never injective. We obtain two fundamental characterizations of universally separably injective spaces: a) A Banach space $E$ is universally separably injective if and only if every separable subspace is contained in a copy of $\ell_\infty$ inside $E$. b) A Banach space $E$ is universally separably injective if and only if for every separable space $S$ one has $\Ext(\ell_\infty/S, E)=0$. The final Section of the paper focuses on special properties of $1$-separably injective spaces. Lindenstrauss\ obtained in the middle sixties a result that can be understood as a proof that, under the continuum hypothesis, $1$-separably injective spaces are $1$-universally separably injective; he left open the question in {\sf ZFC}. We construct a consistent example of a Banach space of type $C(K)$ which is $1$-separably injective but not $1$-universally separably injective.

Archive classification: math.FA

Mathematics Subject Classification: 46A22, 46B04, 46B08, 46A22, 46B04, 46B08, 46B26

Submitted from: castillo@unex.es

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1103.6064

or

http://arXiv.org/abs/1103.6064