Commutator ideal in nonunital C*-algebra Let $A$ be a C*-algebra that has no one-dimensional irreducible representations, that is, there is no (closed) two-sided ideal $I\subseteq A$ such that $A/I\cong\mathbb{C}$.
Let $J$ denote the (not necessarily closed) two-sided ideal generated by additive commutators in $A$:
$$
J:=\{ \sum_{k=1}^n a_k[b_k,c_k]d_k : a_k,b_k,c_k,c_k\in A\}.
$$

Question: Is $A=J$?

The answer is `Yes', if $A$ is unital and in some other cases, but does it hold in general? Note that $J$ is a dense, two-sided ideal (thus contains the Pedersen ideal of $A$), and that $A/J$ is a commutative algebra.
 A: The answer is NO. Rordam and Robert MR3072284 have found
a sequence $(A_n)_n$ of simple unital infinite dimensional C*-algebras
such that $\prod A_n$ has a nonzero character.
(Thanks are due to Yasuhiko Sato for informing me of this.)
Thus the following is true: For every $m$ and $C>1$, there is $n=n(m,C)$
such that  $1=\sum_{k=1}^m a_k[b_k,c_k]d_k$ in $A_n$ implies
$\sum_{k=1}^m\|a_k\| \|b_k\| \|c_k\| \|d_k\| > C$.
Now consider the $c_0$-sum $A:=\bigoplus_m A_{n(m,m^2)}$.
Then $(m^{-1})_m \in A$ cannot be expressed as
$\sum_{k=1}^l a_k[b_k,c_k]d_k$ in $A$, because it would imply
$\sum_{k=1}^l \|a_k(m)\| \|b_k(m)\| \|c_k(m)\| \|d_k(m)\| \geq m$ for every $m\geq l$.
On the the hand, it is easy to show that there are $m$ and $C>1$
that satisfies the following:
For every von Neumann algebra without nonzero abelian direct summand,
one has $1=\sum_{k=1}^m a_k[b_k,c_k]d_k$ for some
$a_k,b_k,c_k,d_k$ with $\sum_{k=1}^m\|a_k\| \|b_k\| \|c_k\| \|d_k\| < C$.
By the Hahn--Banach separation theorem, this implies the following:
For every $A$ without nonzero characters and every $x\in A$,
there are infinite sequences $a_k,b_k,c_k,d_k$ such that
$\sum_{k=1}^\infty\|a_k\| \|b_k\| \|c_k\| \|d_k\| \le C\|x\|$
and $x = \sum_{k=1}^\infty a_k[b_k,c_k]d_k$.
