General solution of corona problem

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.