Corona Theorem

TL;DR

This paper proves a version of the Corona Theorem for a broad class of domains in complex spaces, establishing the density of their canonical images in the spectrum of bounded analytic functions.

Contribution

It extends the Corona Theorem to complex domains like balls and polydisks using abstract algebraic and measure-theoretic methods, including properties of hyper-Stonean spaces and Gleason parts.

Findings

Density of canonical images in the spectrum for various domains

Application of measure and algebraic properties to solve the Gleason problem

Uniform bounds for operators in bidual algebras

Abstract

For a wide class of domains $G\subset\mathbb C^d$ including balls and polydisks we prove the density of their canonical image in the spectrum of $H^\infty(G)$. This Corona Theorem is proved first in its abstract version for certain uniform algebras. We use properties of bands of measures and idempotents corresponding to Gleason parts. The essential tools are properties of hyper-Stonean spaces, normal and Henkin measures and some ideas based on Hoffman - Rossi theorem. We also use our previous results on weak-star closures of reducing bands of measures. Uniform bounds for operators used to solve the Gleason problem concerning ideals of analytic functions vanishing at given points are applied for bidual algebras.