p-adic functionals on torsion-free abelian groups

TL;DR

This paper develops a p-adic duality theory for torsion-free abelian groups, establishing analogues of classical theorems and linking the p-adic double dual to the pro-p completion, with applications to matrix descriptions.

Contribution

It introduces a p-adic dual group framework for torsion-free abelian groups, proves an analogue of Hahn--Banach, and relates the p-adic double dual to the pro-p completion, extending the understanding of these groups.

Findings

P-adic functionals form a rich dual group with an analogue of Hahn--Banach.

The image of G in its p-adic double dual is dense in an appropriate topology.

The p-adic double dual of G equals its pro-p completion.

Abstract

Let p be a prime. A p-adic functional on a torsion-free abelian group G is a group homomorphism from G to the p-adic integers. The group of all such p-adic functionals is viewed as a p-adic dual group of G, and is studied from the point of view of functional analysis. An analogue of the Hahn--Banach Theorem is proved; this result shows that there are sufficiently many p-adic functionals to be interesting. There is a natural homomorphism from G to its p-adic double dual, and one main result that is proved is that the image of G in this double dual is dense in an appropriate topology. This is used to prove the second main result, which says that the p-adic double dual of G is the same as the pro-p completion of G. The theory of p-adic functionals can then be used to produce a matrix description of G if G has finite rank. This matrix description is related to another matrix description due to Malcev.