Grothendieck topology of $C^*$-algebras

TL;DR

This paper introduces a Grothendieck topology for $C^*$-algebras, extending sheaf cohomology to noncommutative spaces, unifying dualities, and generalizing classical topological and algebraic theories.

Contribution

It defines a new Grothendieck topology for $C^*$-algebras, enabling noncommutative sheaf cohomology and unifying various duality theories.

Findings

Provides a noncommutative generalization of sheaf cohomology.

Unifies Gelfand duality with von Neumann algebra duality.

Extends Dixmier-Douady theory to $C^*$-algebras of foliations.

Abstract

For any topological space there is a sheaf cohomology. A Grothendieck topology is a generalization of the classical topology such that it also possesses a sheaf cohomology. On the other hand any noncommutative $C^*$-algebra is a generalization of a locally compact Hausdorff space. Here we define a Grothendieck topology arising from $C^*$-algebras which is a generalization of the topology of the spectra of commutative $C^*$-algebras. This construction yields a noncommutative generalization of the sheaf cohomology of topological spaces. The presented here theory gives a unified approach to the Gelfand duality and the duality between the commutative von Neumann algebras and measure locales. The generalization of the Dixmier-Douady theory concerning $C^*$-algebras of foliations is also discussed.