# Commutators and square-zero elements in C*-algebras

Von | April 14, 2024

Given elements and in a ring, the element is called an (additive) commutator. Given a C*-algebra , we use to denote the additive subgroup (equivalently, the linear subspace) generated by the set of additive commutators in . Note that is not necessarily a closed subspace. Given additive subgroups and , it is customary to use to denote the additive subgroup generated by the set .

An element in a ring is a square-zero element if . Given a square-zero element in a C*-algebra , we consider the polar decomposition in the bidual . Then and belong to , and we have

More generally, Robert showed in Lemma 2.1 in ​[1]​ that every nilpotent element in belongs to . For , we use to denote the set of -nilpotent elements in , and we use to denote the additive subgroup of generated by . We thus have . This raises the following questions:

Question 1: (Robert, Question 2.5 in ​[1]​) Is ?

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

In Theorem 4.2 in ​[2]​, it is shown that , 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 positive answer to Question 1 entails a positive answer to Question 2. Indeed, if a C*-algebra satisfies , then

1. [1]
L. Robert, On the Lie ideals of -algebras, J. Operator Theory (2016) 387–408. https://doi.org/10.7900/jot.2015may17.2070.
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.