Consistent maps and their associated dual representation theorems

Abstract

A 2009 article of Allcock and Vaaler examined the vector space $\mathcal G := \overline{\mathbb Q}^\times/\overline{\mathbb Q}^\times_{\mathrm{tors}}$ 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 $\overline{\mathbb Q}^\times/{\overline{\mathbb Z}}^\times$. In the present article, we use consistent maps to provide representation theorems for the duals of locally constant function spaces on the places of $\overline{\mathbb Q}$ 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.