Let $A$ and $B$ be $C^*$-algebras with ceners $Z_A$ and $Z_B$. Suppose $\rho：A\rightarrow B$ is a surjective $*$- homomorphism. It is easy to check $\rho(Z_A)\subset Z(B)$.

I wonder how to assure that $\rho(Z_A) \neq Z(B)$?

If $rho$ is not surjective, what is the relationship between $\rho(Z_A)$ and $Z_B$?