Structural Hierarchy of Reid Class of non-Archimedean Banach Spaces

Abstract

Let $k$ be a complete valuation field. We formulate a class $\mathscr{R}$ of Banach $k$-vector spaces analogous to Reid class of Abelian groups. We formulate an analogue of the hierarchy of Reid class introduced by K.\ Eda, and verify a counterpart of the classification theorem of Reid class by K.\ Eda. As an application, we verify that the Banach $\mathbb{C}_p$-vector spaces \begin{eqnarray*} & & \ell^{\infty}(\mathbb{N},\mathbb{C}_p), \text{\rm C}_0(\mathbb{N},\mathbb{C}_p), \ell^{\infty}(\mathbb{N},\text{\rm C}_0(\mathbb{N},\mathbb{C}_p)), \text{\rm C}_0(\mathbb{N},\ell^{\infty}(\mathbb{N},\mathbb{C}_p)), \\ & & \ell^{\infty}(\mathbb{N},\text{\rm C}_0(\mathbb{N},\ell^{\infty}(\mathbb{N},\mathbb{C}_p))), \text{\rm C}_0(\mathbb{N},\ell^{\infty}(\mathbb{N},\text{\rm C}_0(\mathbb{N},\mathbb{C}_p))), \end{eqnarray*} and so on are all distinct, the Banach $\mathbb{C}_p$-vector space of bounded continuous functions $\mathbb{Q} \to \mathbb{C}_p$ and its dual Banach $\mathbb{C}_p$-vector spaces cannot be expressed by iterated application of bounded direct product and completed direct sum, and there is no left adjoint functor of the forgetful functor from $\mathscr{R}$ to the category of Banach $\mathbb{C}_p$-vector spaces.