Algebraic topology of $C^*$-algebras
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917343717752832 |
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| author | Ivankov, Petr |
| author_facet | Ivankov, Petr |
| contents | Any $C^*$-algebra can be regarded as a generalization of locally compact, Hausdorff topological space $\mathcal X$. From the commutative commutative Gelfand-Na\uımark theorem it follows that the spectrum of any commutative $C^*$-algebra is a locally compact, Hausdorff space which have the exact information of the $C^*$-algebra. Here we consider a Gelfand spaces of $C^*$-algebras which can be regarded as a generalization of the spectrum. In case of commutative $C^*$-algebras the Gelfand space coincides with the spectrum. Generally Gelfand spaces are not Hausdorff and provide more detailed information of noncommutative $C^*$-algebras. Sometimes the Gelfand space contains the full information of noncommutative $C^*$-algebra. Usage of Gelfand spaces of $C^*$-algebra enables us to define some $C^*$-algebraic analogs of several notions of the classical algebraic topology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_09189 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algebraic topology of $C^*$-algebras Ivankov, Petr Operator Algebras Algebraic Geometry Algebraic Topology Quantum Algebra 46L85 (Primary) 46L35, 46L85, 46L89, 55N30 (Secondary) Any $C^*$-algebra can be regarded as a generalization of locally compact, Hausdorff topological space $\mathcal X$. From the commutative commutative Gelfand-Na\uımark theorem it follows that the spectrum of any commutative $C^*$-algebra is a locally compact, Hausdorff space which have the exact information of the $C^*$-algebra. Here we consider a Gelfand spaces of $C^*$-algebras which can be regarded as a generalization of the spectrum. In case of commutative $C^*$-algebras the Gelfand space coincides with the spectrum. Generally Gelfand spaces are not Hausdorff and provide more detailed information of noncommutative $C^*$-algebras. Sometimes the Gelfand space contains the full information of noncommutative $C^*$-algebra. Usage of Gelfand spaces of $C^*$-algebra enables us to define some $C^*$-algebraic analogs of several notions of the classical algebraic topology. |
| title | Algebraic topology of $C^*$-algebras |
| topic | Operator Algebras Algebraic Geometry Algebraic Topology Quantum Algebra 46L85 (Primary) 46L35, 46L85, 46L89, 55N30 (Secondary) |
| url | https://arxiv.org/abs/2511.09189 |