Corona theorem

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In mathematics, the corona theorem is a result in complex analysis and Banach algebra theory. It was proved in 1962 by Lennart Carleson in his article Interpolations by bounded analytic functions and the corona problem.

It concerns the commutative Banach algebra and Hardy space H^∞ of bounded holomorphic functions on the open unit disc D. Its spectrum contains for each z in D a point P_z, corresponding to the maximal ideal of functions f with

f(z) = 0.

The points P_z cannot make up the entire spectrum S, essentially because the spectrum is a compact space and D is not. The corona theorem states that they make up a dense subset of S; or in other words S is a compactification of D, and contains no 'corona' points lying in the complement of the closure of D.

Carleson showed that this question reduced rather quickly to more classical formulations, on finite sets of analytic functions taking small values, and reminiscent of the Nullstellensatz. Proofs are combinatorial in nature.

Cole later showed that this result cannot be extended to all open Riemann surfaces (Reference: LMS notes of Gamelin).