Let $\mathcal{A}$ be a $C^{\ast}$-algebra. A subset $S$ of $A$ is called multiplicatively closed if $ 0 \notin S$ and $ab \in S$ for all $a, b \in S$.
There exists a closed prime ideal $P$ of $\mathcal{A}$ such that $ P \cap S = \phi$
Let $\mathcal{F}$ be the collection of ideals of $\mathcal{A}$ satisfying $P \cap S = \phi.$ By applying Zorn's Lemma to $\mathcal{F}$ , I'm able to get a prime ideal $P$ which satisfies $ P \cap S = \phi$ but i ain't able to get closed prime ideal. Any hints or ideas?
You won't be able to in general since closed prime ideals of a C*-algebra are always maximal.
If $A$ is abelian, this isn't too hard to see. Let $P$ be a closed prime ideal of $A$. Then $A/P$ is $C_0(X)$ for some locally compact Hausdorff space $X$ and is an integral domain. But this can only happen if $X$ is a single point, in which case $A/P$ is the field $\mathbb{C}$ and $P$ is thus maximal.