Is the center $Z$ of a C*-algebra $A$ unital if and only if $A$ is unital?

Question

Let $A$ be a C*-algebra. Its center is defined by $$Z:=\{z\in A:az=za \ \text{for all} \ a\in A\}.$$ It is easy to verify that $Z$ is a C*-subalgebra of $A$. Also, if $A$ is unital, then it is clear that $1_{A}\in A$. But is the converse also true? In other words, is $A$ unital if $Z$ has a unit $1_{Z}$? I know that this is not true for general C*-subalgebras $B$ of $A$, but I cannot think of any counterexamples in the specific case where $B$ is the center $Z$ of some C*-algebra $A$.

Answer 1

If $H$ is an infinite-dimensional Hilbert space, then the algebra $K(H)$ of compact operators on $H$ is non-unital, but its center is just the trivial algebra which is unital. If you want an example where the center is nontrivial, you could take $K(H)\times\mathbb{C}$.