Rings and C*-algebras generated by commutators

TL;DR

This paper characterizes when rings and C*-algebras are generated by their commutators, providing bounds on the number of products needed and exploring implications for various classes of algebras.

Contribution

It establishes precise conditions and bounds for rings and C*-algebras to be generated by commutators, including new results for matrix rings and Jiang-Su-stable C*-algebras.

Findings

Matrix rings require at most 2 products of commutators.

C*-algebras with certain properties need at most 3 or 6 products.

Every element in the commutator ideal has a power that is a sum of products of commutators.

Abstract

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.