Revista Matemática Iberoamericana


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Volume 31, Issue 4, 2015, pp. 1167–1230
DOI: 10.4171/RMI/866

Published online: 2015-12-23

Towards Oka–Cartan theory for algebras of holomorphic functions on coverings of Stein manifolds. II

Alexander Brudnyi[1] and Damir Kinzebulatov[2]

(1) University of Calgary, Canada
(2) Research in Mathematical Sciences, Toronto, Canada

We establish basic results of complex function theory within certain algebras of holomorphic functions on coverings of Stein manifolds (such as algebras of Bohr’s holomorphic almost periodic functions on tube domains or algebras of all fibrewise bounded holomorphic functions arising, e.g., in the corona problem for $H^{\infty}$). In particular, in this context we obtain results on holomorphic extension from complex sub manifolds, properties of divisors, corona-type theorems, holomorphic analogues of the Peter–Weyl approximation theorem, Hartogs-type theorems, characterizations of uniqueness sets, etc. Our proofs are based on analogues of Cartan theorems A and B for coherent-type sheaves on maximal ideal spaces of these algebras proved in Part I.

Keywords: Oka–Cartan theory, algebras of holomorphic functions, coverings of complex manifolds

Brudnyi Alexander, Kinzebulatov Damir: Towards Oka–Cartan theory for algebras of holomorphic functions on coverings of Stein manifolds. II. Rev. Mat. Iberoamericana 31 (2015), 1167-1230. doi: 10.4171/RMI/866