Opuscula Math. 41, no. 6 (2021), 843-848
https://doi.org/10.7494/OpMath.2021.41.6.843
Opuscula Mathematica
Corona theorem for strictly pseudoconvex domains
Abstract. Nearly 60 years have passed since Lennart Carleson gave his proof of Corona Theorem for unit disc in the complex plane. It was only recently that M. Kosiek and K. Rudol obtained the first positive result for Corona Theorem in multidimensional case. Using duality methods for uniform algebras the authors proved "abstract" Corona Theorem which allowed to solve Corona Problem for a wide class of regular domains. In this paper we expand Corona Theorem to strictly pseudoconvex domains with smooth boundaries.
Keywords: Corona theorem, Banach algebra, uniform algebra, Arens product, Gleason part, band of measures, representing measure.
Mathematics Subject Classification: 30H80, 32A38, 32A65, 32A70, 46J10, 46J15, 46J20.
- L. Carleson, Interpolation by bounded analytic functions and the corona problem, Ann. of Math. 76 (1962), 547-559.
- B.J. Cole, R.M. Range, A-measures on complex manifolds and some applications, J. Funct. Anal. 11 (1972), 393-400.
- H.G. Dales, F.K. Dashiell, Jr., A.T.-M. Lau, D. Strauss, Banach Spaces of Continuous Functions as Dual Spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer-Verlag, New York, 2016.
- T.W. Gamelin, Uniform Algebras, Prentice Hall, Inc., Englewood Clifs, N.J. 1969.
- M. Hakim, N. Sibony, Spectre de \(A(\overline\Omega)\) pour les domains borneés faiblement pseudoconvexes réguliers, J. Funct. Anal. 37 (1980), 127-135.
- M. Kosiek, K. Rudol, Dual algebras and \(A\)-measures, J. Funct. Spaces 2014 2014), Article ID 364271.
- M. Kosiek, K. Rudol, Corona Theorem, arXiv:2106.15683.
- A. Noell, Peak points for pseudoconvex domains: A survey, J Geom. Anal. 18 (2008), 1058-1087.
- H. Rossi, Holomorphically convex sets in several complex variables, Ann. Math. 74 (1961), 470-493.
- Sebastian Gwizdek
https://orcid.org/0000-0002-9149-8507- AGH University of Science and Technology, Faculty of Applied Mathematics AGH, al. Mickiewicza 30, 30-059 Kraków, Poland
- Communicated by P.A. Cojuhari.
- Received: 2021-07-01.
- Revised: 2021-08-12.
- Accepted: 2021-08-13.
- Published online: 2021-11-29.
