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| Main Author: | |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2211.14684 |
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| _version_ | 1866915085671202816 |
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| author | Izzo, Alexander J. |
| author_facet | Izzo, Alexander J. |
| contents | It is shown that there exists a compact planar set K such that the uniform algebra R(K) is nontrivial and strongly regular. This settles an issue raised by Donald Wilken 55 years ago. It is shown that the set K can be chosen such that, in addition, R(K) is not weakly amenable. It is also shown that there exists a uniform algebra that has bounded relative units but is not weakly amenable. These results answer questions raised by Joel Feinstein and Matthew Heath 17 years ago. A key ingredient in our proofs is a bound we establish on the functions introduced by Thomas Koerner to simplify Robert McKissick's construction of a nontrivial normal uniform algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_14684 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A sharper Swiss cheese Izzo, Alexander J. Complex Variables Functional Analysis It is shown that there exists a compact planar set K such that the uniform algebra R(K) is nontrivial and strongly regular. This settles an issue raised by Donald Wilken 55 years ago. It is shown that the set K can be chosen such that, in addition, R(K) is not weakly amenable. It is also shown that there exists a uniform algebra that has bounded relative units but is not weakly amenable. These results answer questions raised by Joel Feinstein and Matthew Heath 17 years ago. A key ingredient in our proofs is a bound we establish on the functions introduced by Thomas Koerner to simplify Robert McKissick's construction of a nontrivial normal uniform algebra. |
| title | A sharper Swiss cheese |
| topic | Complex Variables Functional Analysis |
| url | https://arxiv.org/abs/2211.14684 |