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]L. Robert, On the Lie ideals of
-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.