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&infin; consists of the bounded holomorphic functions on the open unit disc D. Its spectrum S (the closed maximal ideals) contains D as an open subspace because for each z in D there is a maximal ideal consisting of functions f with


 * f(z) = 0.

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
 * $$|f_1|+\cdots+|f_n|\ge\delta$$ everywhere in the unit ball.

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In 1979 Thomas Wolff gave a simplified (but unpublished) proof of the corona theorem, described in and.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 introduced 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 assumes the continuity up to the boundary in the corona theorem, then the conclusion follows easily from the theory of commutative Banach algebra.