  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 submanifolds, 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.
