Commutators and square-zero elements in C*-algebras

Von | April 14, 2024

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. [1]
    L. Robert, On the Lie ideals of \Cstar-algebras, J. Operator Theory (2016) 387–408.
  2. [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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