Consistent maps and their associated dual representation theorems

TL;DR

The paper develops a Riesz-type representation framework for duals of spaces of locally constant functions on places of $\overline{\mathbb Q}$ by introducing consistent maps and the spaces $\mathcal{J}_S^*$ and $\mathcal{J}_S'$, linking them to $LC_c(Y_S)^*$ and $LC_c(Y_S)'$ through explicit isomorphisms $\Phi_S^*$ and $\Phi_S'$. It establishes algebraic and continuous dual representations for general sets of places $S$, identifies the codimension-1 kernels with $\operatorname{span}\{\lambda\}$, and demonstrates density results for $\mathcal{F}_S=\{f_{S,\alpha}\}$ in the relevant function spaces. The work unifies and extends previous results of Allcock–Vaaler and Samuels, and generalizes the framework to arbitrary number fields $F$, showing that the notion of consistency is essentially field-independent. The resulting dual-representation theorems provide a robust arithmetic analogue of the classical Riesz representation, with potential applications to further dualities and generalizations in the study of height functions and place-wise function spaces.

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.

Theorems & Definitions (20)

• Proposition 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 1.4
• Theorem 1.5
• Theorem 1.6
• proof : Proof of Proposition \ref{['CompactnessRestriction']}
• Lemma 2.1
• proof
• proof : Proof of Theorem \ref{['MainIsomorphism']}