Universal covering space of the noncommutative torus

TL;DR

This paper introduces a simple example of a noncommutative universal covering space related to the noncommutative torus, contributing to the development of noncommutative topology without complex algebraic prerequisites.

Contribution

It provides an accessible example of a noncommutative universal covering space, advancing the understanding of noncommutative topological invariants.

Findings

Sample of noncommutative universal covering space presented

The example is easy to understand and does not require Hopf-Galois extensions

Contributes to the broader theory of noncommutative universal coverings

Abstract

Gelfand - Na\u{i}mark 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.