Duality of analytic Hopf algebras and the Amice transform

TL;DR

This paper develops a general framework for analytic Hopf algebras over arbitrary Banach rings, extending $p$-adic Fourier theory and Amice duality to a broader, prime-independent setting.

Contribution

It generalizes K"othe spaces to arbitrary Banach rings and constructs global analytic Hopf algebras with duality properties, unifying $p$-adic and non-$p$-adic cases.

Findings

Constructed global analytic Hopf algebras over general Banach rings.

Proved reflexivity and nuclearity of these spaces.

Established a prime-independent version of Amice duality.

Abstract

We construct global versions of the analytic Hopf algebras used in the $p$-adic Fourier theory of Amice and Mahler over a general Banach ring, independently of the choice of prime $p$. This is done by generalising K\"othe echelon and coechelon spaces to an arbitrary base Banach ring $R$ and proving reflexivity and nuclearity results. We show how to define an analytic Hopf algebra structure on them and investigate their duality theory. The particular case of the Hopf algebra of analytic functions converging on the open unit disk around $1$ and its dual is studied in detail. Amice duality is recovered from this case by base-change to a $p$-adic ring. Most notably, when $R$ is the ring of integers with the trivial norm, we obtain a global analytic version of Amice duality that does not depend on $p$.