Consistent maps and their associated dual representation theorems

A 2009 article of Allcock and Vaaler examined the vector space $\mathcal{G}:={\bar{\mathbb{Q}}}^{\times }\mathrm{/}{\bar{\mathbb{Q}}}_{\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}}^{\times }$ over $\mathbb{Q}$, describing its completion with respect to the Weil height as a certain $L^1$ space. By involving an object called a consistent map, the author began efforts to establish Riesz-type representation theorems for the duals of spaces related to $\mathcal{G}$. Specifically, we provided such results for the algebraic and continuous duals of ${\bar{\mathbb{Q}}}^{\times }\mathrm{/}{\bar{\mathbb{Z}}}^{\times }$. In the present article, we use consistent maps to provide representation theorems for the duals of locally constant function spaces on the places of $\bar{\mathbb{Q}}$ that arise in the work of Allcock and Vaaler. We further apply our new results to recover, as a corollary, a main theorem of our previous work.