Products of commutators in simple algebras
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Soggetti: | |
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| _version_ | 1866914062177140736 |
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| author | Brešar, Matej Jang, Hau-Yuan Robert, Leonel |
| author_facet | Brešar, Matej Jang, Hau-Yuan Robert, Leonel |
| contents | Let $A$ be a finite-dimensional simple algebra that is not a field. We show that every $a\in A$ can be written as
$a=(bc-cb)(de-ed)$ for some
$b,c,d,e\in A$. This is not always true for infinite-dimensional simple algebras. In fact, for any $m\in \mathbb N$ we provide an example of an infinite-dimensional simple unital $C^*$-algebra $A$ in which $1$ cannot be written as $\sum_{i=1}^m x_i(a_ib_i-b_ia_i)y_i$ for some $x_i,a_i,b_i,y_i\in A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_23956 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Products of commutators in simple algebras Brešar, Matej Jang, Hau-Yuan Robert, Leonel Rings and Algebras Let $A$ be a finite-dimensional simple algebra that is not a field. We show that every $a\in A$ can be written as $a=(bc-cb)(de-ed)$ for some $b,c,d,e\in A$. This is not always true for infinite-dimensional simple algebras. In fact, for any $m\in \mathbb N$ we provide an example of an infinite-dimensional simple unital $C^*$-algebra $A$ in which $1$ cannot be written as $\sum_{i=1}^m x_i(a_ib_i-b_ia_i)y_i$ for some $x_i,a_i,b_i,y_i\in A$. |
| title | Products of commutators in simple algebras |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2509.23956 |