  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.
