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Consistent maps and their associated dual representation theorems

Volume 178 / 2025

Charles L. Samuels Colloquium Mathematicum 178 (2025), 11-30 MSC: Primary 08C20, 11R04, 32C37; Secondary 11G50, 46B10, 46B25 DOI: 10.4064/cm9180-3-2025 Published online: 7 April 2025

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.

Authors

  • Charles L. SamuelsDepartment of Mathematics
    Christopher Newport University
    Newport News, VA 23606, USA
    e-mail

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