Abstract: We develop the basic elements of complex function theory within certain subalgebras of holomorphic functions on coverings of complex manifolds (including holomorphic extension from complex submanifolds, properties of divisors, corona type theorem, holomorphic analogue of Peter-Weyl approximation theorem, Hartogs type theorem, characterization of the uniqueness sets, etc). Our model examples are: (1) subalgebra of Bohr's holomorphic almost periodic functions on tube domains (i.e. the uniform limits of exponential polynomials) (2) subalgebra of all fibrewise bounded holomorphic functions (arising in corona problem for $H^\infty$) (3) subalgebra of holomorphic functions having fibrewise limits.

Our proofs are based on the analogues of Cartan theorems A and B for coherent type sheaves on the maximal ideal spaces of these subalgebras.

This is joint work with Alexander Brudnyi.