General solution of corona problem

TL;DR

The work develops an abstract corona framework for uniform algebras by exploiting the bidual $A^{**}$, bands of measures, and Gleason parts to connect the spectra of $A^{**}$-type algebras with $H^{\infty}$-type algebras. It shows that, under a fiber-singleton and norming conditions, the canonical image of a Gleason part $G$ is dense in the spectrum of $H^{\infty}(A,\mathcal{M}_G)$, thereby solving corona-type questions for a broad class of domains, including balls and polydisks, through a unifying measure-theoretic approach. The results yield concrete corona theorems for domains such as strictly pseudoconvex regions with $C^2$ boundaries and strongly star-like cartesian products, via isomorphisms with $H^{\infty}(G)$ and stability of $G$. Overall, the paper integrates uniform algebra theory, Arens products, and spectral analysis to extend corona results to higher dimensions and provide a robust framework for density of canonical images in spectra. The approach has potential implications for complex function theory in several variables and operator-algebraic formulations of function algebras on complex domains.

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.

Theorems & Definitions (51)

• Theorem 1.1: Corona Theorem
• Definition 2.1
• Proposition 2.2
• proof
• Lemma 2.3
• Theorem 2.4
• Definition 3.1
• Lemma 3.2
• proof
• Remark 4.1