Grothendieck topology of $C^*$-algebras
Petr R. Ivankov

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 -algebra is a generalization of a locally compact Hausdorff space. Here we define a Grothendieck topology arising from -algebras which is a generalization of the topology of the spectra of commutative -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 -algebras of foliations is also discussed.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Advanced Banach Space Theory · Mathematical Analysis and Transform Methods
