Quantization of noncompact coverings

A rigorous algebraic definition of noncommutative coverings is developed. In the case of commutative algebras, this definition is equivalent to the classical definition of topological coverings of locally compact spaces. The theory has the following nontrivial applications:
- Coverings of the quantum SU(2) group,
- Coverings of isospectral deformations of Spin – manifolds,
- Coverings of noncommutative tori.
The theory supplies the rigorous definition of noncommutative Wilson lines.