Products of commutators in simple algebras

Matej Brešar; Hau-Yuan Jang; Leonel Robert

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arXiv · 2509.23956

Products of commutators in simple algebras

Matej Brešar·Hau-Yuan Jang·Leonel Robert

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Abstract

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 $x_i,a_i,b_i,y_i\in A$.

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BibTeX RIS

Matej Brešar, Hau-Yuan Jang, Leonel Robert. 2025-09-28. Products of commutators in simple algebras. https://arxiv.org/abs/2509.23956

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