Is there a $C^{*}$ algebra with these properties

Question

Is there a unital C* algebra A which is NOT simple but satisfies the following two conditions:

1)A has trivial center

2)A has a faithful trace such that every zero trace element lies in the closure of the span of commutator elements.

By a commutator element I mean an element in the form xy-yx.

For a related MO post see this question

Answer 1

Yes and you don't even have to take the closure. For instance if $A$ is a non-simple C*-algebra with trivial cetnre that has unique (faithful) trace (for instance $A=C^*(G)$ for a sufficiently non-commutative amenable group such as the group of permutations of integers that move at most finitely many entries), then $A\otimes \mathcal{Z}$, where $\mathcal{Z}$ is the Jiang-Su algebra, has the property that each zero-trace element is a sum of 7 commutators. This is a nice result of Ng and Robert.

Actually for most nice C*-algebras we have $A\otimes \mathcal{Z}\cong A$.

The result concerning taking the closure is easier and it does not involve the Jiang-Su algebra. See Robert's answer here.