论文标题
电晕定理的另一个证明
Another proof of the corona theorem
论文作者
论文摘要
令$ h^\ infty（δ）$为开放单元圆盘$δ$上有限分析函数的统一代数，让$ \ mathfrak {m} {m}（h^\ infty）$是$ h^\ infty（δ）$的最大理想空间。通过将$δ$作为$ \ mathfrak {m}（h^\ infty）$的开放子集，Corona问题询问$Δ$在$ \ Mathfrak {M}（H^\ Infty）$中是否密集，这是由L. Carleson肯定地解决的。将集群价值定理扩展到有限的许多函数的情况下，我们提供了Corona定理的直接证明：让$ ϕ $成为$ \ Mathfrak {M} {M}（H^\ infty）$中的同构，让$ f_1，f_1，f_2，f_2，f_2，\ dots，f_n $在$ h^^$ h^f_n $中正常功能。然后有一个序列$ \ {ζ_j\} $ in $δ$满足$ f_k（ζ_j）\ rightarrow ϕ（f_k）$ for $ k = 1，2，\ dots，n $。另一方面，电晕问题在许多一般环境中仍未解决，例如某些平面域，多盘和球，我们的方法是如此自然，以至于可以从另一种角度来处理此类情况。
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.
