Bounded linear endomorphisms of rigid analytic functions

TL;DR

This paper investigates the relationship between sheaves of differential operators and bounded endomorphisms on rigid analytic spaces over non-Archimedean fields, establishing conditions for their isomorphism.

Contribution

It proves that the natural map between the sheaf of infinite order differential operators and bounded endomorphisms is an isomorphism only under specific algebraic conditions on the ground field.

Findings

The isomorphism holds if and only if the field is algebraically closed with an uncountable residue field.

The result generalizes known complex-analytic cases to non-Archimedean rigid analytic spaces.

Provides criteria for when differential operators fully capture bounded endomorphisms.

Abstract

Let $K$ be a field of characteristic zero complete with respect to a non-trivial, non-Archimedean valuation. We relate the sheaf $\widehat{\mathcal{D}}$ of infinite order differential operators on smooth rigid $K$-analytic spaces to the algebra $\mathcal{E}$ of bounded $K$-linear endomorphisms of the structure sheaf. In the case of complex manifolds, Ishimura proved that the analogous sheaves are isomorphic. In the rigid analytic situation, we prove that the natural map $\widehat{\mathcal{D}} \to \mathcal{E}$ is an isomorphism if and only if the ground field $K$ is algebraically closed and its residue field is uncountable.