Another proof of the corona theorem

Abstract

Let $H^\infty(\Delta)$ be the uniform algebra of bounded analytic functions on the open unit disc $\Delta$, and let $\mathfrak{M}(H^\infty)$ be the maximal ideal space of $H^\infty(\Delta)$. By regarding $\Delta$ as an open subset of $\mathfrak{M}(H^\infty)$, the corona problem asks whether $\Delta$ 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 $\phi$ be a homomorphism in $\mathfrak{M}(H^\infty)$, and let $f_1, f_2, \dots, f_N$ be functions in $H^\infty(\Delta)$. Then there is a sequence $\{\zeta_j\}$ in $\Delta$ satisfying$f_k(\zeta_j) \rightarrow \phi(f_k)$ 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.