# 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 Kakutani (1941) and proved by LennartCarleson  ( 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,...,fn generate the unit ideal of H if and only if there is some δ>0 such that

${\displaystyle |f_{1}|+\cdots +|f_{n}|\geq \delta }$ 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In 1979 Thomas Wolff gave a simplified (but unpublished) proof of the corona theorem, described in ( Koosis 1980 ) and ( Gamelin 1980 ).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.

Note that if one assumes the continuity up to the boundary in Corona's theorem, then the conclusion follows easily from the theory of commutative Banach algebra ( Rudin 1991 ).

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## 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:, MR   0599306, Zbl   0466.46050
• Kakutani, Shizuo (1941). "Concrete representation of abstract (M)-spaces. (A characterization of the space of continuous functions.)". Ann. of Math. Series 2. 42 (4): 994–1024. doi:10.2307/1968778. hdl:. JSTOR   1968778. MR   0005778.
• 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
• Rudin, Walter (1991), Functional Analysis, p. 279.
• 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 .