Rings and C*-algebras generated by commutators
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866910396814721024 |
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| author | Gardella, Eusebio Thiel, Hannes |
| author_facet | Gardella, Eusebio Thiel, Hannes |
| contents | We show that a unital ring is generated by its commutators as an ideal if and only if there exists a natural number $N$ such that every element is a sum of $N$ products of pairs of commutators. We show that one can take $N \leq 2$ for matrix rings, and that one may choose $N \leq 3$ for rings that contain a direct sum of matrix rings -- this in particular applies to C*-algebras that are properly infinite or have real rank zero. For Jiang-Su-stable C*-algebras, we show that $N\leq 6$ can be arranged.
For arbitrary rings, we show that every element in the commutator ideal admits a power that is a sum of products of commutators. We prove that a C*-algebra cannot be a radical extension over a proper ideal, and we use this to deduce that a C*-algebra is generated by its commutators as a not necessarily closed ideal if and only if every element is a finite sum of products of pairs of commutators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_05958 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rings and C*-algebras generated by commutators Gardella, Eusebio Thiel, Hannes Rings and Algebras Operator Algebras Primary 16N60, 46L05, 47B47. Secondary 15A15, 15A23, 16S50, 19K14 We show that a unital ring is generated by its commutators as an ideal if and only if there exists a natural number $N$ such that every element is a sum of $N$ products of pairs of commutators. We show that one can take $N \leq 2$ for matrix rings, and that one may choose $N \leq 3$ for rings that contain a direct sum of matrix rings -- this in particular applies to C*-algebras that are properly infinite or have real rank zero. For Jiang-Su-stable C*-algebras, we show that $N\leq 6$ can be arranged. For arbitrary rings, we show that every element in the commutator ideal admits a power that is a sum of products of commutators. We prove that a C*-algebra cannot be a radical extension over a proper ideal, and we use this to deduce that a C*-algebra is generated by its commutators as a not necessarily closed ideal if and only if every element is a finite sum of products of pairs of commutators. |
| title | Rings and C*-algebras generated by commutators |
| topic | Rings and Algebras Operator Algebras Primary 16N60, 46L05, 47B47. Secondary 15A15, 15A23, 16S50, 19K14 |
| url | https://arxiv.org/abs/2301.05958 |