Non-Archimedean Countably Injective Banach Spaces

作者：Perez Garcia C^*

来源：Journal of Convex Analysis, 2015, 22(3): 733-746.

• Non-Archimedean Banach spaces
• injective spaces
• orthonormal bases
• spaces of continuous functions

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 R or C), recently studied in [2]. 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, [6], [7], obtained that, under the continuum hypothesis, 1-separably injective Banach spaces over R or 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 [2], 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 (Theorem 3.3). Additionally, for spaces of continuous functions on a zero-dimensional compact space, we get the coincidence of all the non-Archimedean injectivity properties (Theorems 4.3, 4.4).

• 出版日期2015