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| In [[mathematics]], the '''corona theorem''' is a result about the [[spectrum of a commutative Banach algebra|spectrum]] of the [[Bounded function|bounded]] [[holomorphic function]]s on the [[open unit disc]], conjectured by {{harvtxt|Kakutani|1941}} and proved by {{harvs|authorlink=Lennart Carleson|first=Lennart|last=Carleson|year=1962|txt=yes}}.
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| The commutative Banach algebra and [[Hardy space]] [[H infinity|''H''<sup>∞</sup>]] consists of the bounded [[holomorphic function]]s on the [[open unit disc]] ''D''. Its [[spectrum of a commutative Banach algebra|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 | |
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| :''f''(''z'') = 0.
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| 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 {{harvtxt|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 ''f''<sub>1</sub>,...,''f''<sub>''n''</sub> generate the unit ideal of ''H''<sup>∞</sup> if and only if there is some δ>0 such that | |
| :<math>|f_1|+\cdots+|f_n|\ge\delta</math> everywhere in the unit ball.
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| Newman showed that the corona theorem can be reduced to an interpolation problem, which was then proved by Carleson.
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| In 1979 [[Thomas Wolff]] gave a simplified (but unpublished) proof of the corona theorem, described in {{harv|Koosis|1980}} and {{harv|Gamelin|1980}}.
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| Cole later showed that this result cannot be extended to all [[open Riemann surface]]s {{harv|Gamelin|1978}}.
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| 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.
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| ==See also==
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| *[[Corona set]]
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| ==References==
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| *{{citation|mr=0141789 | zbl = 0112.29702
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| |last= Carleson
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| |first= Lennart
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| |author-link= Lennart Carleson
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| |title=Interpolations by bounded analytic functions and the corona problem
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| |journal= [[Annals of Mathematics]]
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| |issue= 2
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| |volume=76
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| |year= 1962
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| |pages=547–559
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| |doi=10.2307/1970375
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| |jstor=1970375
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| }}
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| *{{citation|mr=0521440 | zbl = 0418.46042
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| |last=Gamelin|first= T. W.
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| |title=Uniform algebras and Jensen measures.
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| |series=London Mathematical Society Lecture Note Series
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| |volume= 32
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| |publisher= [[Cambridge University Press]]
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| |place= Cambridge-New York
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| |year= 1978
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| |pages= iii+162
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| |isbn= 978-0-521-22280-8}}
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| *{{citation|mr=0599306 | zbl = 0466.46050
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| |last=Gamelin|first= T. W.
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| |title=Wolff's proof of the corona theorem
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| |journal=[http://www.ma.huji.ac.il/~ijmath/ Israel Journal of Mathematics]
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| |volume= 37
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| |year=1980
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| |issue= 1–2
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| |pages= 113–119
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| |doi=10.1007/BF02762872}}
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| *{{citation|mr=0565451 | zbl = 0435.30001
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| |last=Koosis|first= Paul
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| |title=Introduction to H<sup>''p''</sup>-spaces. With an appendix on Wolff's proof of the corona theorem
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| |series=London Mathematical Society Lecture Note Series
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| |volume= 40
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| |publisher= [[Cambridge University Press]]
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| |place= Cambridge-New York
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| |year=1980
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| |pages= xv+376
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| |isbn= 0-521-23159-0}}
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| *{{citation|mr=0106290 | zbl = 0092.11802
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| |last= Newman
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| |first= D. J.
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| |title= Some remarks on the maximal ideal structure of H<sup>∞</sup>
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| |journal= [[Annals of Mathematics]]
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| |issue= 2
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| |volume= 70
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| |year= 1959
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| |pages= 438–445
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| |doi=10.2307/1970324
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| |jstor=1970324
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| }}
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| *{{citation
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| |mr=0125442 | zbl = 0139.30402
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| |last=Schark
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| |first= I. J.
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| |title=Maximal ideals in an algebra of bounded analytic functions
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| |journal=[[Indiana University Mathematics Journal|Journal of Mathematics and Mechanics]]
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| |volume= 10
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| |url=http://www.iumj.indiana.edu/IUMJ/FULLTEXT/1961/10/10050
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| |year=1961
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| |pages=735–746}}.
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| [[Category:Banach algebras]]
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| [[Category:Hardy spaces]]
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| [[Category:Theorems in complex analysis]]
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