
Journal of Convex Analysis 22 (2015), No. 3, 733746 Copyright Heldermann Verlag 2015 NonArchimedean Countably Injective Banach Spaces Cristina PerezGarcia Dept. of Mathematics, Faculty of Sciences, Universidad de Cantabria, Avda. de los Castros s/n, 39071 Santander, Spain perezmc@unican.es The main purpose of this paper is to investigate the relationships between some classes of nonArchimedean injective Banach spaces. The results obtained reveal sharp and interesting contrasts with the classical situation (i.e. for Banach spaces over the reals R or the complex numbers C, recently studied by A. Avilés, F. Cabello Sánchez, J.M.F. Castillo, M. González and Y. Moreno [On separably injective Banach spaces, Adv. Math. 234 (2013) 192216]. One of those contrasts has to do with a classical open problem whose roots come back to 1964. In fact, in that year J. Lindenstrauss [Extension of compact operators, Mem. Amer. Math. Soc. 48 (1964); On the extension of operators with range in a C(K) space, Proc. Amer. Math. Soc. 15 (1964) 218225] obtained that, under the continuum hypothesis, 1separably injective Banach spaces over R or C are 1universally separably injective. He left open the question in the usual setting of set theory with the Axiom of Choice. A negative answer, for a Banach space of continuous functions on a compact space, was given in the first paper cited above, where the authors also posed a so natural classical problem as the following one: Without the continuum hypothesis, 1separably injective classical Banach spaces must be universally separably injective? However, we prove in this paper that, for any nonArchimedean Banach space, all the 1injectivity properties coincide. Additionally, for spaces of continuous functions on a zerodimensional compact space, we get the coincidence of all the nonArchimedean injectivity properties. Keywords: NonArchimedean Banach spaces, injective spaces, orthonormal bases, spaces of continuous functions. MSC: 46S10 [ Fulltextpdf (158 KB)] for subscribers only. 