Non-Archimedean Countably Injective Banach Spaces

The main purpose of this paper is to investigate the relationships between some classes of non-Archimedean injective Banach spaces. The results obtained reveal sharp and interesting contrasts with the classical situation (i.e. for Banach spaces over the reals $\mathbb{R}$ or the complex numbers $\mathbb{C}$, 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) 192-216]. 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) 218-225] obtained that, under the continuum hypothesis, 1-separably injective Banach spaces over $\mathbb{R}$ or $\mathbb{C}$ are 1-universally 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, 1-separably injective classical Banach spaces must be universally separably injective? However, we prove in this paper that, for any non-Archimedean Banach space, all the 1-injectivity properties coincide. Additionally, for spaces of continuous functions on a zero-dimensional compact space, we get the coincidence of all the non-Archimedean injectivity properties.