# Corona theorem

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,...,fn generate the unit ideal of H∞ if and only if there is some δ>0 such that {stile di visualizzazione |f_{1}|+cdot +|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In 1979 Thomas Wolff gave a simplified (but unpublished) proof of the corona theorem, described in (Koosis 1980) e (Gamelin 1980).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 (Rudino 1991).

See also Corona set References Carleson, Lennart (1962), "Interpolations by bounded analytic functions and the corona problem", Annali di matematica, 76 (3): 547–559, doi:10.2307/1970375, JSTOR 1970375, SIG 0141789, Zbl 0112.29702 Gamelin, T. w. (1978), Uniform algebras and Jensen measures., Serie di appunti per le lezioni della London Mathematical Society, vol. 32, Cambridge-New York: Cambridge University Press, pp. iii+162, ISBN 978-0-521-22280-8, SIG 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, SIG 0599306, Zbl 0466.46050 Kakutani, Shizuo (1941). "Concrete representation of abstract (M)-spaces. (A characterization of the space of continuous functions.)". Anna. di matematica. Serie 2. 42 (4): 994–1024. doi:10.2307/1968778. hdl:10338.dmlcz/100940. JSTOR 1968778. SIG 0005778. Koosis, Paolo (1980), Introduction to Hp-spaces. With an appendix on Wolff's proof of the corona theorem, Serie di appunti per le lezioni della London Mathematical Society, vol. 40, Cambridge-New York: Cambridge University Press, pp. xv+376, ISBN 0-521-23159-0, SIG 0565451, Zbl 0435.30001 Uomo nuovo, D. J. (1959), "Some remarks on the maximal ideal structure of H∞", Annali di matematica, 70 (2): 438–445, doi:10.2307/1970324, JSTOR 1970324, SIG 0106290, Zbl 0092.11802 Rudino, Walter (1991), Analisi funzionale, p. 279. Schark, io. J. (1961), "Maximal ideals in an algebra of bounded analytic functions", Journal of Mathematics and Mechanics, 10: 735–746, SIG 0125442, Zbl 0139.30402. nascondi vte Analisi funzionale (argomenti – glossario) Spazi BanachBesovFréchetHilbertHölderNucleareOrliczSchwartzSobolevVettore topologico Proprietà barrelledcompletatodual (algebrico/topologico)localmente convessoriflessivoseparabile TeoremiHahn–BanachRieszrappresentazionegrafo chiusoprincipio di limitatezza uniformeKakutani punto fissoKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operatori adjointboundedcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebres Algebra di BanachC*-algebraspettro di un'algebra C*problemi di un operatore algebra localmente compatto di un'algebra di Neumanngruppo compatto di un'algebra di Neumann Problema del sottospazio Congettura di Mahler Applicazioni Spazio di Hardy Teoria spettrale delle equazioni differenziali ordinarie Heat Kernel Teorema dell'indice Calcolo delle variazioni Calcolo funzionale Operatore integrale Polinomio di Jones Teoria dei campi quantistici topologici Geometria non commutativa Ipotesi di Riemann Distribuzione (o funzioni generalizzate) Argomenti avanzati proprietà di approssimazione insieme bilanciato Teoria di Choquet topologia debole Distanza di Banach–Mazur Teoria di Tomita–Takesaki Categorie: Banach algebrasHardy spacesTheorems in complex analysis

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