A nontrivial uniform algebra Dirichlet on its maximal ideal space
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914213187813376 |
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| author | Izzo, Alexander J. |
| author_facet | Izzo, Alexander J. |
| contents | It is shown that there exists a nontrivial uniform algebra that is Dirichlet on its maximal ideal space and has a dense set of elements that are exponentials. This answers a 65-year-old question of John Wermer and a 17-year-old question of Garth Dales and Joel Feinstein. Our example is P(X) for a certain compact set X in complex Euclidean 2-space ($\mathbb{C}^2$). It is also shown that there exists a logmodular uniform algebra with proper Shilov boundary but with no nontrivial Gleason parts. This answers a modification of another 65-year-old question of Wermer. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_19583 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A nontrivial uniform algebra Dirichlet on its maximal ideal space Izzo, Alexander J. Complex Variables Functional Analysis Primary 32A38, 32E30, 46J10, 46J15, Secondary 30H50, 32A65, 32E20 It is shown that there exists a nontrivial uniform algebra that is Dirichlet on its maximal ideal space and has a dense set of elements that are exponentials. This answers a 65-year-old question of John Wermer and a 17-year-old question of Garth Dales and Joel Feinstein. Our example is P(X) for a certain compact set X in complex Euclidean 2-space ($\mathbb{C}^2$). It is also shown that there exists a logmodular uniform algebra with proper Shilov boundary but with no nontrivial Gleason parts. This answers a modification of another 65-year-old question of Wermer. |
| title | A nontrivial uniform algebra Dirichlet on its maximal ideal space |
| topic | Complex Variables Functional Analysis Primary 32A38, 32E30, 46J10, 46J15, Secondary 30H50, 32A65, 32E20 |
| url | https://arxiv.org/abs/2403.19583 |