Hull kernel topology and $\tau_w$-topology are same on Prime$(A)$

Question

Let $A$ be a $C^{\ast}$-algebra and Id$(A)$ denotes the space of ideals of $A$. A subbasis for $\tau_w$-topology is given by the collection $$U(J)= \{ I \in Id(A): I \nsupseteq J \}, J \in Id(A)$$

Let Prime$(A)$ denotes the space of prime ideals of $A$. Equip Prime$(A)$ with hull kernel topology.

Is it true that restriction of $\tau_w$-topology on Prime$(A)$ is same as hull kernel topology?

Any ideas or references?