Quasifolds, diffeology and noncommutative geometry

Abstract

After embedding the objects quasifolds into the category $\{$Diffeology$\}$, we associate a ${\mathbf{C}}^{\ast }$-algebra with every atlas of any quasifold, and show how different atlases give Morita equivalent algebras. This builds a new bridge between diffeology and noncommutative geometry – beginning with the today classical example of the irrational torus – which associates a Morita class of ${\mathbf{C}}^{\ast }$-algebras with a diffeomorphic class of quasifolds.