In the study of commutators. The important question is whether the linear span of commutators [A, A] is closed in norm topology. In particular, we shall discuss the case that algebra is unital and simple. The following question is naturally raised.
Let $A$ be an infinite dimensional unital simple Banach algebra. Let $[A, A]$ denote the linear span of commutators in $A$, where a commutator in $A$ is an element of the form $xy-yx$, $x, y \in A$. We say that $A$ has property $\mathbb{X}$ if $A \neq [A, A]$ and $A=\overline{[A, A]}$, where $\overline{[A, A]}$ denotes the norm closure of $[A, A]$.
$\textbf{Question}$:
Which infinite dimensional unital simple Banach algebras $A$ have properties $A \neq [A, A]$ and $A=\overline{[A, A]}$?
We note that the question has no solution if A is a C*-algebra. This is due to C. Pop's results.