Consistent maps and their associated dual representation theorems
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866916694983704576 |
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| author | Samuels, Charles L. |
| author_facet | Samuels, Charles L. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_12887 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Consistent maps and their associated dual representation theorems Samuels, Charles L. Number Theory Functional Analysis Primary 08C20, 11R04, 32C37, Secondary 11G50, 46B10, 46B25 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. |
| title | Consistent maps and their associated dual representation theorems |
| topic | Number Theory Functional Analysis Primary 08C20, 11R04, 32C37, Secondary 11G50, 46B10, 46B25 |
| url | https://arxiv.org/abs/2306.12887 |