Universal covering space of the noncommutative torus

Abstract

Gelfand - Naĭmark theorem supplies contravariant functor from a category of commutative $C^*-$ algebras to a category of locally compact Hausdorff spaces. Therefore any commutative $C^*-$ algebra is an alternative representation of a topological space. Similarly a category of (noncommutative) $C^*-$ algebras can be regarded as a category of generalized (noncommutative) locally compact Hausdorff spaces. Generalizations of topological invariants may be defined by algebraic methods. For example Serre Swan theorem states that complex topological $K$ - theory coincides with $K$ - theory of $C^*$ - algebras. However the algebraic topology have a rich set of invariants. Some invariants do not have noncommutative generalizations yet. This article contains a sample of noncommutative universal covering. General theory of noncommutative universal coverings is being developed by the author of this article. However this sample has independent interest, it is very easy to understand and does not require knowledge of Hopf-Galois extensions.