Grothendieck topology of $C^*$-algebras
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914732512903168 |
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| author | Ivankov, Petr R. |
| author_facet | Ivankov, Petr R. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_14684 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Grothendieck topology of $C^*$-algebras Ivankov, Petr R. Operator Algebras Algebraic Geometry 46L85 (Primary) 46L80, 55N30, 55N05, 55N15, 18F10, 18F20 (Secondary) 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. |
| title | Grothendieck topology of $C^*$-algebras |
| topic | Operator Algebras Algebraic Geometry 46L85 (Primary) 46L80, 55N30, 55N05, 55N15, 18F10, 18F20 (Secondary) |
| url | https://arxiv.org/abs/2306.14684 |