Hausdorff blowups of $C^*$-algebra spectra and their applications

In algebraic geometry, there is a well-known blowup procedure, which is a surjection of a nonsingular variety onto a singular one. This operation is very useful due to numerous nice properties of nonsingular varieties. Similarly, there is a notion of a "Hausdorff blowup", which is a mapping of a Hausdorff space onto the spectrum of a $C^*$-algebra. Many results related to the spectrum also hold for their Hausdorff blowups. For example, a noncommutative covering space of a $C^*$-algebra with a Hausdorff spectrum corresponds to a topological covering space of the spectrum, or to a Hausdorff blowup. This yields a unified viewpoint to covering spaces of $C^*$-algebras with a Hausdorff spectrum and groupoid $C^*$-algebras.