Another proof of the corona theorem

TL;DR

This paper offers a new, direct proof of the corona theorem for bounded analytic functions on the unit disc, extending the cluster value theorem to finitely many functions and discussing potential applications to other domains.

Contribution

It provides a novel, straightforward proof of the corona theorem using an extension of the cluster value theorem to multiple functions, with implications for broader settings.

Findings

Confirmed the density of the disc in the maximal ideal space for the unit disc

Extended the cluster value theorem to finitely many functions

Proposed a natural approach potentially applicable to other domains

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.