Grothendieck topology of $C^{\ast }$-algebras

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 $C^{\ast }$-algebra is a generalization of a locally compact Hausdorff space. Here we define a Grothendieck topology arising from $C^{\ast }$-algebras which is a generalization of the topology of the spectra of commutative $C^{\ast }$-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 $C^{\ast }$-algebras of foliations is also discussed.

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