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∈A. We say A has property X if A≠[A, A] and A equals the norm closure of [A, A].

Which infinite dimensional unital simple Banach algebras A have properties A≠[A, A] and A equals the norm closure of [A, A]?

We note that the question has no solution if A is a C*-algebra. This is due to C. Pop's results.