Corona theorem

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In mathematics, the corona theorem is a result about the spectrum of the bounded holomorphic functions on the open unit disc, conjectured by Schark (1961) 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 In 1979 Wolff gave a simplified (but unpublished) proof of the corona theorem, described in (Koosis 1980) and (Gamelin 1980).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).

Famous open question: corona theorem for every plain domain. Also there are not known examples corona theorem for multidimensional domains.

[edit] References

• Carleson, Lennart (1962), “Interpolations by bounded analytic functions and the corona problem”, Ann. of Math. (2) 76: 547--559, MR0141789, <http://links.jstor.org/sici?sici=0003-486X%28196211%292%3A76%3A3%3C547%3AIBBAFA%3E2.0.CO%3B2-5> 
• Gamelin, T. W. (1978), Uniform algebras and Jensen measures., vol. 32, London Mathematical Society Lecture Note Series, Cambridge-New York: Cambridge University Press, pp. iii+162, MR0521440 , ISBN 978-0521222808 
• Gamelin, T. W. (1980), “Wolff's proof of the corona theorem.”, Israel J. Math. 37 (1-2): 113-119, MR0599306 
• Koosis, Paul (1980), Introduction to H^p-spaces. With an appendix on Wolff's proof of the corona theorem., vol. 40, London Mathematical Society Lecture Note Series, Cambridge-New York: Cambridge University Press, pp. xv+376, MR0565451, ISBN 0-521-23159-0 
• Newman, D. J. (1959), “Some remarks on the maximal ideal structure of H^∞”, Ann. of Math. (2) 70: 438-445, MR0106290, <http://links.jstor.org/sici?sici=0003-486X%28195911%292%3A70%3A3%3C438%3ASROTMI%3E2.0.CO%3B2-K> 
• Schark, I. J. (1961), “Maximal ideals in an algebra of bounded analytic functions.”, J. Math. Mech. 10: 735-746, MR0125442