A classification of $\mathbb{Q}$-linear maps from ${\bar{\mathbb{Q}}}^{\times }\mathrm{/}{\bar{\mathbb{Q}}}_{\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}}^{\times }$ to $\mathbb{R}$

A 2009 article of Allcock and Vaaler explored the $\mathbb{Q}$-vector space $\mathcal{G}:={\bar{\mathbb{Q}}}^{\times }\mathrm{/}{\bar{\mathbb{Q}}}_{\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}}^{\times }$, showing how to represent it as part of a function space on the places of $\bar{\mathbb{Q}}$. 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.