Resumen de Towards Oka–Cartan theory for algebras of holomorphic functions on coverings of Stein manifolds I

Alexander Brudnyi, Damir Kinzebulatov

• 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∞).

  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.