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.