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

$|f_{1}| \cdots |f_{n}|\geq \delta$ 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In 1979 Thomas Wolff gave a simplified (but unpublished) proof of the corona theorem, described in (Koosis 1980) and (Gamelin 1980).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 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 (Rudin 1991).

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, vol. 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
• 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:10338.dmlcz/100940. JSTOR 1968778. MR 0005778.
• Koosis, Paul (1980), Introduction to H^p-spaces. With an appendix on Wolff's proof of the corona theorem, London Mathematical Society Lecture Note Series, vol. 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.