A classification of $\mathbb Q$-linear maps from $\overline{\mathbb Q}^\times/\overline{\mathbb Q}^\times_{\mathrm{tors}}$ to $\mathbb R$
Charles L. Samuels
TL;DR
The paper resolves the structure of the $\mathbb{Q}$-linear dual of the Archimedean-places representation of the multiplicative group modulo torsion, by proving a representation theorem: the map $\Phi^*:\mathcal{J}^*\to \mathcal{L}(\mathcal{G},\mathbb{R})$ is surjective with $\ker(\Phi^*)=\mathrm{span}_\mathbb{R}\{\lambda\}$. This yields an isomorphism $\mathcal{J}_q^*\cong \mathcal{L}(\mathcal{G},\mathbb{R})$ for suitable $F$ and place $q$, enabling classification of extensions of completely additive arithmetic functions to $\mathcal{G}$. The work combines the Allcock–Vaaler functional-analytic model with Dirichlet's $S$-unit theorem and a canonical construction of consistent maps to describe the full landscape of $\mathbb{Q}$-linear maps from $\mathcal{G}$ to $\mathbb{R}$ and, via the KRational framework, to identify rational-valued extensions on $K^\times$. Overall, the results connect arithmetic function extensions with functionals on a place-space, providing a robust algebraic-dual description and concrete mechanisms to build rational-valued functionals.
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.