closed ideal in a $C^*$- algebra

Question

Suppose $A$ is a non-simple $C^*$-algebra, let $x_0$ be a nonzero element in $A$, and let $S=\{x_0y-yx_0:y\in A\}$. If $I$ is the closed ideal generated by the set $S$. I think there is a possibility that $I=A$, but I cannot think of a concrete example.

Answer 1

If $A$ is a simple $C^*$-algebra with trivial center, then the ideal $I$ as you constructed must be the whole algebra. For we must have $I=A$ or $\{0\}$, and if $I=\{0\}$, then $x_0$ is in the center of $A$, in which case $x_0=0$.

An example of such an algebra is $K(H)$ for a separable infinite-dimensional Hilbert space $H$.

If you want the additional assumption that $A$ is non-simple, let $A=K(H)\oplus K(H)$, and let $x_0=(x_1,x_1)$ for any nonzero element $x_1\in K(H)$.