Commutators and square-zero elements in C*-algebras

Given elements $b$ and $c$ in a ring, the element $[b,c] := bc-cb$ is called an (additive) commutator. Given a C*-algebra $A$ , we use $[A,A]$ to denote the additive subgroup (equivalently, the linear subspace) generated by the set of additive commutators in $A$ . Note that $[A,A]$ is not necessarily a closed subspace. Given additive subgroups $V$ and $W$ , it is customary to use $[V,W]$ to denote the additive subgroup generated by the set $\{[v,w] : v\in V, w \in W\}$ .

An element $a$ in a ring is a square-zero element if $a^2=0$ . Given a square-zero element $x$ in a C*-algebra $A$ , we consider the polar decomposition $x=v|x|$ in the bidual $A^{**}$ . Then $|x|^{1/2}$ and $|x|^{1/2}v$ belong to $A$ , and we have

$x = [|x|^{1/2},|x|^{1/2}v] \in [A,A].$

More generally, Robert showed in Lemma 2.1 in [1] that every nilpotent element in $A$ belongs to $[A,A]$ . For $k\geq 2$ , we use $N_k(A)$ to denote the set of $k$ -nilpotent elements in $A$ , and we use $N_k(A)^+$ to denote the additive subgroup of $A$ generated by $N_k(A)$ . We thus have $N_k(A)^+ \subseteq [A,A]$ . This raises the following questions:

Question 1: (Robert, Question 2.5 in [1]) Is $[A,A] = N_2(A)^+$ ?

A positive answer to the above question is known in many cases, in particular if $A$ is unital and admits no characters (one-dimensional irreducible representations), by Theorem 4.2 in [1]. Further, it is known that $[A,A]$ is always contained in the closure of $N_2(A)^+$ .

In Theorem 4.2 in [2], it is shown that $N_2(A)^+ = [N_2(A)^+,N_2(A)^+]$ , that is, every square-zero element is a sum of commutators of square-zero elements, and every commutator of square-zero elements is a sum of square-zero elements. This raises the closely related question if every commutator in a C*-algebra is a sum of commutators of commutators:

Question 2: (Question 3.5 in [2]) Is $[A,A]=[[A,A],[A,A]]$ ?

A positive answer to Question 1 entails a positive answer to Question 2. Indeed, if a C*-algebra $A$ satisfies $[A,A]=N_2(A)^+$ , then

$[[A,A],[A,A]] = [N_2(A)^+,N_2(A)^+] = N_2(A)^+ = [A,A].$

[1]
L. Robert, On the Lie ideals of $\Cstar$ -algebras, J. Operator Theory (2016) 387–408. https://doi.org/10.7900/jot.2015may17.2070.
[2]
E. Gardella, H. Thiel, Prime ideals in C*-algebras and applications to Lie theory, Proc. Amer. Math. Soc. (to Appear) 1 (2024) 9.

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