Inverse Limits of Noncommutative Covering Projections

TL;DR

This paper develops an algebraic framework for inverse limits of noncommutative covering projections and demonstrates that Moyal planes can be viewed as inverse limits of noncommutative tori.

Contribution

It introduces a pure algebraical construction of inverse limits in noncommutative covering projections, extending topological concepts to noncommutative geometry.

Findings

Moyal planes are inverse limits of noncommutative tori

Provides a new algebraic approach to noncommutative inverse limits

Extends topological invariants to noncommutative settings

Abstract

The Gelfand - Na\u{i}mark theorem supplies the one to one correspondence between commutative $C^*$-algebras and locally compact Hausdorff spaces. So any noncommutative $C^*$-algebra can be regarded as a generalization of a topological space. Generalizations of several topological invariants can be defined by algebraical methods. This article contains a pure algebraical construction of inverse limits in the category of (noncommutative) covering projections. It is proven that Moyal planes are inverse limits of covering projections of noncommutative tori.