General solution of corona problem

Abstract

Using a description of the spectrum of bidual algebra $A^{\ast \ast }$ 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 $A^{\ast \ast }$. 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.

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