Another proof of the corona theorem
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866929392459972608 |
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| author | Tanaka, Jun-ichi |
| author_facet | Tanaka, Jun-ichi |
| contents | Let $H^\infty(Δ)$ be the uniform algebra of bounded analytic functions on the open unit disc $Δ$, and let $\mathfrak{M}(H^\infty)$ be the maximal ideal space of $H^\infty(Δ)$. By regarding $Δ$ as an open subset of $\mathfrak{M}(H^\infty)$, the corona problem asks whether $Δ$ is dense in $\mathfrak{M}(H^\infty)$, which was solved affirmatively by L. Carleson. Extending the cluster value theorem to the case of finitely many functions, we provide a direct proof of the corona theorem: Let $ϕ$ be a homomorphism in $\mathfrak{M}(H^\infty)$, and let $f_1, f_2, \dots, f_N$ be functions in $H^\infty(Δ)$. Then there is a sequence $\{ζ_j\}$ in $Δ$ satisfying$f_k(ζ_j) \rightarrow ϕ(f_k)$ for $k=1, 2, \dots, N$. On the other hand, the corona problem remains unsolved in many general settings, for instance, certain plane domains, polydiscs and balls, our approach is so natural that it may be possible to deal with such cases from another point of view. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_10126 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Another proof of the corona theorem Tanaka, Jun-ichi Complex Variables Functional Analysis Primary 30H80, Secondary 30H10, 46J10, 46J20, 37A46 Let $H^\infty(Δ)$ be the uniform algebra of bounded analytic functions on the open unit disc $Δ$, and let $\mathfrak{M}(H^\infty)$ be the maximal ideal space of $H^\infty(Δ)$. By regarding $Δ$ as an open subset of $\mathfrak{M}(H^\infty)$, the corona problem asks whether $Δ$ is dense in $\mathfrak{M}(H^\infty)$, which was solved affirmatively by L. Carleson. Extending the cluster value theorem to the case of finitely many functions, we provide a direct proof of the corona theorem: Let $ϕ$ be a homomorphism in $\mathfrak{M}(H^\infty)$, and let $f_1, f_2, \dots, f_N$ be functions in $H^\infty(Δ)$. Then there is a sequence $\{ζ_j\}$ in $Δ$ satisfying$f_k(ζ_j) \rightarrow ϕ(f_k)$ for $k=1, 2, \dots, N$. On the other hand, the corona problem remains unsolved in many general settings, for instance, certain plane domains, polydiscs and balls, our approach is so natural that it may be possible to deal with such cases from another point of view. |
| title | Another proof of the corona theorem |
| topic | Complex Variables Functional Analysis Primary 30H80, Secondary 30H10, 46J10, 46J20, 37A46 |
| url | https://arxiv.org/abs/2204.10126 |