Let $A$ be a $C^{\ast}$-algebra and $I$ be an ideal of $A$. Recall that $I$ is called semiprime if $I=A$ or $I$ is intersection of prime ideals. In this paper, it is mentioned without proof that
Closed ideals of $A$ are semiprime ideals.
I have not seen proof of this. Can someone give me any reference for proof of this?
Closed ideals of a $C^*$-algebra $A$ are not only semiprime but semiprimitive. Primitive ideals are precisely the kernels of irreducible $*$-representations of $A$, thus every primitive ideal is prime.
Step 1: Let $R$ be the set of the irreducible GNS representations of $A$. In any textbook about $C^*$-algebras, you can find that $R$ separates the points of $A$. Thus, $\bigcap\{\ker\pi: \pi\in R\} = \{0\}$. Consequently, any $C^*$-algebra is semiprimitive, so semiprime.
Step 2: Let $I$ be a closed ideal of $A$. Consider the $C^*$-algebra $B=A/I$. Let $\phi:A\to A/I$ be the quotient homomorphism.
Let $R$ be the set of irreducible GNS representations of $B$. Let $R_I = \{\pi\circ\phi: \pi\in R\}$. Since $$\bigcap_{\pi\in R} \pi^{-1}(\{0\}) = \bigcap_{\pi\in R}\ker\pi = \{I\}$$ by step 1, then $$\bigcap_{\gamma\in R_I}\ker\gamma = \bigcap_{\pi\in R}\ker(\pi\circ\phi) = \phi^{-1}\left(\bigcap_{\pi\in R} \pi^{-1}(\{0\}) \right) = \phi^{-1}(\{I\}) = I.$$
edit: 1. Martin Argerami kindly pointed out that an explanation is missing/necessary about why $\ker\pi$ is a prime ideal for an irreducible representation $\pi:A\to B(H)$. A proof of this can be found in Theorem 4.1.8 (b) in Palmer's book. Alternatively, for $a,b\in A$ suppose $aAb\subseteq\ker\pi$. We want to show either $\pi(a) = 0$ or $\pi(b)=0$.
For a contradiction, suppose $\exists a,b\in A$ such that $aAb\subseteq\ker\pi$, but $\pi(a) \neq 0$ and $\pi(b)\neq 0$. Since $\pi(a)\neq 0$, then $\ker(\pi(a))\neq H$. Since $\pi(b)\neq 0$, there exists $\xi\in H$ such that $\eta:=\pi(b)\xi\neq 0$. Consequently, $\pi(A)\eta = \{\pi(x)\eta:x\in A\}\subseteq\ker(\pi(a))\neq H$. This contradicts the fact that $\pi$ is irreducible.
2. This bullet point is not a part of the answer but related. It's been known that every closed prime ideal in a separable $C^*$-algebra is the kernel of some irreducible representation. However, the same is not true for non-separable $C^*$-algebras. The first known example of a $C^*$-algebra that contains a non-primitive closed prime ideal was given by Nik Weaver to the best of my knowledge in the beginning of 2000s. There has since been other examples, some of which are graph $C^*$-algebras. Sorry for the missing references at the given moment, which I can provide at a later time if asked.