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- W2600979453 abstract "In this paper we deal with two weaker forms of injectivity which turn out 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 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 spaces: a) A Banach space $E$ is universally separably 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 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 spaces. Lindenstrauss obtained in the middle sixties a result that can be understood as a proof that, under the continuum hypothesis, 1-separably 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 but not 1-universally separably injective. We show that, under the continuum hypothesis, to be universally separably injective is not a $3$-space property, as we wrongly claimed in the paper mentioned in the title." @default.
- W2600979453 created "2017-04-07" @default.
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- W2600979453 date "2011-03-30" @default.
- W2600979453 modified "2023-09-27" @default.
- W2600979453 title "On separably injective Banach spaces and Corrigendum to On separably injective Banach spaces [Adv. Math. 234 (2013) 192--216]" @default.
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