Towards Oka-cartan Theory for Algebras of Fibrewise Bounded Holomorphic Functions on Coverings of Stein Manifolds I

We develop complex function theory within certain algebras of holomorphic functions on coverings of Stein manifolds. This, in particular, includes the topics of holomorphic extension from complex submanifolds, corona type theorems, properties of divisors, holomorphic analogs of the Peter-Weyl approximation theorem, Hartogs type theorems, characterization of uniqueness sets. Our model examples covered by this theory are: (1) algebra of Bohr’s holomorphic almost periodic functions on tube domains; (2) algebra of all fibrewise bounded holomorphic functions (e.g., arising in the corona problem for H∞). Our basic tools used in the proofs are new Cartan type theorems A and B for analogs of coherent sheaves on the maximal ideal spaces of these algebras generalizing the classical Cartan theorems.