  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 representation theorems for the duals of spaces
related to $\mathcal G$. Specifically, we provided representation theorems for
the algebraic and continuous duals of $\overline{\mathbb
Q}^\times/{\overline{\mathbb Z}}^\times$. We explore further applications of
consistent maps to provide representation theorems for duals of other spaces
arising in the work of Allcock and Vaaler. In particular, we establish a
connection between consistent maps and locally constant functions on the space
of places of $\overline{\mathbb Q}$. We further apply our new results to
recover, as a corollary, a main theorem of our previous work.
