A note on Pontryagin duality and continuous logic

TL;DR

This paper demonstrates that Pontryagin duality can be viewed as a special case of Stone duality within a continuous logic framework, linking topological groups, their duals, and the Bohr compactification.

Contribution

It establishes a novel connection between Pontryagin duality and Stone duality through continuous logic, providing a new perspective on duality theories for abelian topological groups.

Findings

Pontryagin duality is a special case of Stone duality in continuous logic.

The local type space corresponds to the Pontryagin dual of the group of homomorphisms.

The local type space is identified with the Bohr compactification of the group.

Abstract

We exhibit Pontryagin duality as a special case of Stone duality in a continuous logic setting. More specifically, given an abelian topological group $A$, and $\mathcal F$ the family (group) of continuous homomorphisms from $A$ to the circle group $\mathbb T$, then, viewing $(A,+)$ equipped with the collection $\mathcal F$ as a continuous logic structure $M$, we show that the local type space $S_\mathcal F(M)$ is precisely the Pontryagin dual of the group $\mathcal F$ where the latter is considered as a discrete group. We conclude, using Pontryagin duality (between compact and discrete abelian groups), that $S_\mathcal F(M)$ is the Bohr compactification of the topological group $A$.