A classification of $\mathbb Q$-linear maps from $\overline{\mathbb Q}^\times/\overline{\mathbb Q}^\times_{\mathrm{tors}}$ to $\mathbb R$

Abstract

A 2009 article of Allcock and Vaaler explored the $\mathbb Q$-vector space $\mathcal G := \overline{\mathbb Q}^\times/{\overline{\mathbb Q}^\times_{\mathrm{tors}}}$, showing how to represent it as part of a function space on the places of $\overline{\mathbb Q}$. We establish a representation theorem for the $\mathbb R$-vector space of $\mathbb Q$-linear maps from $\mathcal G$ to $\mathbb R$, enabling us to classify extensions to $\mathcal G$ of completely additive arithmetic functions. We further outline a strategy to construct $\mathbb Q$-linear maps from $\mathcal G$ to $\mathbb Q$, i.e., elements of the algebraic dual of $\mathcal G$. Our results make heavy use of Dirichlet's $S$-unit Theorem as well as a measure-like object called a consistent map, first introduced by the author in previous work.