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 Kakutani (1941) and proved by Lennart Carleson (1962).

The commutative Banach algebra and Hardy space H^∞ 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 Newman (1959), 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,...,f_n generate the unit ideal of H^∞ if and only if there is some δ>0 such that

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In 1979 Thomas Wolff gave a simplified (but unpublished) proof of the corona theorem, described in (Koosis 1980) and (Gamelin 1980).

Cole later showed that this result cannot be extended to all open Riemann surfaces (Gamelin 1978).

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.

See also

• Corona set

References

• Carleson, Lennart (1962), "Interpolations by bounded analytic functions and the corona problem", Annals of Mathematics 76 (3): 547–559, doi:10.2307/1970375, JSTOR 1970375, MR 0141789, Zbl 0112.29702 
• Gamelin, T. W. (1978), Uniform algebras and Jensen measures., London Mathematical Society Lecture Note Series 32, Cambridge-New York: Cambridge University Press, pp. iii+162, ISBN 978-0-521-22280-8, MR 0521440, Zbl 0418.46042 
• Gamelin, T. W. (1980), "Wolff's proof of the corona theorem", Israel Journal of Mathematics 37 (1–2): 113–119, doi:10.1007/BF02762872, MR 0599306, Zbl 0466.46050 
• Koosis, Paul (1980), Introduction to Hp-spaces. With an appendix on Wolff's proof of the corona theorem, London Mathematical Society Lecture Note Series 40, Cambridge-New York: Cambridge University Press, pp. xv+376, ISBN 0-521-23159-0, MR 0565451, Zbl 0435.30001 
• Newman, D. J. (1959), "Some remarks on the maximal ideal structure of H^∞", Annals of Mathematics 70 (2): 438–445, doi:10.2307/1970324, JSTOR 1970324, MR 0106290, Zbl 0092.11802 
• Schark, I. J. (1961), "Maximal ideals in an algebra of bounded analytic functions", Journal of Mathematics and Mechanics 10: 735–746, MR 0125442, Zbl 0139.30402 .