Towards Oka-Cartan theory for algebras of holomorphic functions on coverings of Stein manifolds II

TL;DR

This paper develops foundational complex function theory for specific algebras of holomorphic functions on coverings of Stein manifolds, extending classical results to new algebraic contexts.

Contribution

It introduces analogues of Cartan theorems A and B for these algebras and proves key properties like extension, divisors, corona theorems, and approximation results.

Findings

Established holomorphic extension from submanifolds

Proved corona type theorems for these algebras

Developed analogues of Cartan theorems A and B

Abstract

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.