The affine group of a local field is Hermitian
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911174928367616 |
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| author | Carter, Max |
| author_facet | Carter, Max |
| contents | The question of whether the group $\mathbb{Q}_p \rtimes \mathbb{Q}_p^*$ is Hermitian has been stated as an open question in multiple sources in the literature, even as recently as a paper by R. Palma published in 2015. In this note we confirm that this group is Hermitian by proving the following more general theorem: given any local field $\mathbb{K}$, the affine group $\mathbb{K} \rtimes \mathbb{K}^*$ is a Hermitian group. The proof is a consequence of results about Hermitian Banach $*$-algebras from the 1970's. In the case that $\mathbb{K}$ is a non-archimedean local field, this result produces examples of totally disconnected locally compact Hermitian groups with exponential growth, and these are the first examples of groups satisfying these properties. This answers a second question of Palma about the existence of such groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_06893 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The affine group of a local field is Hermitian Carter, Max Functional Analysis Group Theory 22D15, 43A20, 43A45, 46H05 The question of whether the group $\mathbb{Q}_p \rtimes \mathbb{Q}_p^*$ is Hermitian has been stated as an open question in multiple sources in the literature, even as recently as a paper by R. Palma published in 2015. In this note we confirm that this group is Hermitian by proving the following more general theorem: given any local field $\mathbb{K}$, the affine group $\mathbb{K} \rtimes \mathbb{K}^*$ is a Hermitian group. The proof is a consequence of results about Hermitian Banach $*$-algebras from the 1970's. In the case that $\mathbb{K}$ is a non-archimedean local field, this result produces examples of totally disconnected locally compact Hermitian groups with exponential growth, and these are the first examples of groups satisfying these properties. This answers a second question of Palma about the existence of such groups. |
| title | The affine group of a local field is Hermitian |
| topic | Functional Analysis Group Theory 22D15, 43A20, 43A45, 46H05 |
| url | https://arxiv.org/abs/2504.06893 |