General solution of corona problem

TL;DR

This paper develops an abstract corona theorem for certain uniform algebras, proving the density of specific Gleason parts and solving the corona problem positively for domains like balls and polydisks.

Contribution

It introduces a new abstract corona theorem based on the spectrum of bidual algebras, extending corona problem solutions to broader classes of domains.

Findings

Density of Gleason parts in the spectrum of $H^$-type subalgebras

Isometric isomorphism between subalgebras and $H^(G)$ for various domains

Positive solutions to the corona problem for balls and polydisks

Abstract

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.