Corona theorem


In mathematics, the corona theorem is a result about the spectrum of the bounded holomorphic functions on the open unit disc, conjectured by and proved by .
The commutative Banach algebra and Hardy space H∞ consists of the bounded holomorphic functions on the open unit disc D. Its spectrum S contains D as an open subspace because for each z in D there is a maximal ideal consisting of functions f with
The subspace D cannot make up the entire spectrum S, essentially because the spectrum is a compact space and D is not. The complement of the closure of D in S was called the corona by, and the corona theorem states that the corona is empty, or in other words the open unit disc D is dense in the spectrum. A more elementary formulation is that elements f1,...,fn generate the unit ideal of H∞ if and only if there is some δ>0 such that
Newman showed that the corona theorem can be reduced to an interpolation problem, which was then proved by Carleson.
In 1979 Thomas Wolff gave a simplified proof of the corona theorem, described in and.
Cole later showed that this result cannot be extended to all open Riemann surfaces.
As a by-product, of Carleson's work, the Carleson measure was invented which itself is a very useful tool in modern function theory. It remains an open question whether there are versions of the corona theorem for every planar domain or for higher-dimensional domains.
Note that if one assume the continuity up to the boundary in Corona's theorem, then the conclusion follows easily from the theory of Commutative Banach algebra.