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

Abstract

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.