General solution of corona problem
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917355761696768 |
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| author | Kosiek, Marek Rudol, Krzysztof |
| author_facet | Kosiek, Marek Rudol, Krzysztof |
| contents | Using a description of the spectrum of bidual algebra $A^{**}$ of a uniform algebra $A$ we obtain abstract corona theorem for certain uniform algebras. It asserts the density of a specific Gleason part in the spectrum of an $H^\infty$ -- type subalgebra of $A^{**}$.
There is an isometric isomorphism of the latter subalgebra with $H^\infty(G)$ for a wide class of domains $G\subset\mathbb C^d$. Using abstract corona theorem we show the density of the canonical image of $G$ in the spectrum of $H^\infty(G)$, solving positively corona problem for such domains. In particular, we obtain positive solution for balls and polydisks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_20220 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | General solution of corona problem Kosiek, Marek Rudol, Krzysztof Functional Analysis Primary 30H80, 32A65, 46J15, Secondary 32A38, 46J20 Using a description of the spectrum of bidual algebra $A^{**}$ of a uniform algebra $A$ we obtain abstract corona theorem for certain uniform algebras. It asserts the density of a specific Gleason part in the spectrum of an $H^\infty$ -- type subalgebra of $A^{**}$. There is an isometric isomorphism of the latter subalgebra with $H^\infty(G)$ for a wide class of domains $G\subset\mathbb C^d$. Using abstract corona theorem we show the density of the canonical image of $G$ in the spectrum of $H^\infty(G)$, solving positively corona problem for such domains. In particular, we obtain positive solution for balls and polydisks. |
| title | General solution of corona problem |
| topic | Functional Analysis Primary 30H80, 32A65, 46J15, Secondary 32A38, 46J20 |
| url | https://arxiv.org/abs/2505.20220 |