Given a simple $C^*$-Algebra $A$, consider its center $C = A \cap A'$, i.e. the set of elements in $A$ commuting with every other element in $A$.
I have shown that if $A$ is unital, its center is trivial, i.e. $C = \mathbb{C} \cdot I$. It should be the case that if $A$ is non-unital, we have $C = 0$, but I don't know how to approach showing this.
In the unital case I considered the spectrum of an arbitrary element $a \in C$, which has to be nonempty, and showed that for some $\lambda \in \sigma(a)$ the set $\overline{(\lambda - c)A}$ is an ideal not containing $1$, so it must be $\{0\}$. However, I cannot do that here.