Coverings of foliation algebras

TL;DR

This paper explores the relationship between topological coverings of foliated manifolds and noncommutative coverings of their operator algebras, highlighting a natural but not bijective correspondence.

Contribution

It introduces a geometric construction linking topological and noncommutative coverings of foliations, clarifying the scope and limitations of this correspondence.

Findings

Established a natural correspondence between topological and noncommutative coverings.

Identified noncommutative coverings that do not fit the geometric construction.

Clarified the non-bijective nature of the relationship.

Abstract

This article is devoted to the geometric construction which states a natural correspondence between topological coverings of a foliated manifolds and noncommutative coverings of the operator algebras. However this correspondence is not one to one because there are noncommutaive coverings of foliations which do not comply with discussed in this article construction.