Let $A$ and $B$ be $C^{\ast}$-algebras with centers $Z$ and $Z'$ respectively. Let $\phi:A \to B$ be surjective $C^{\ast}$-morphism. Then
$\phi(Z)=Z'$ if and only if the map $\Phi: \operatorname{Prim}(Z') \to \operatorname{Prim}(Z)$ defined as $\Phi(J) = \phi^{-1}(J)$ is injective.
Can someone please give me reference for the above result?
The above result is mentioned without proof in Proposition $1$ of the paper titled On the homomorphic image of Center of $C^{\ast}$-algebras by Vesterstrom.
P.S: The same was asked on mathoverflow too but unfortunately dint get any answer. Copy of the same question on mathoverflow can be found here.