Towards Oka–Cartan theory for algebras of 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 results on 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. The model examples of these algebras are: (1) Bohr's algebra of holomorphic almost periodic functions on tube domains; (2) algebra of all fibrewise bounded holomorphic functions (e.g., arising in the corona problem for $H^\infty$). Our approach is based on an extension of the classical Oka-Cartan theory to coherent-type sheaves on the maximal ideal spaces of these algebras -- topological spaces having some features of complex manifolds.