Let $A$ be a unital C*-algebra. For each pure state $\tau$ of $A$ defines $N_\tau:=\{a\in A|\tau(a^*a)=0\}$, and denote the collection of all these $N_\tau$ by $\mathcal N$. For each closed left ideal $L$ of $A$, defines $$\vee L:=\{N_\tau\in\mathcal N|L\subset N_\tau\}.$$ Then $\vee A=\emptyset$ and $\vee \{0\}=\mathcal N$, and $\cap_i\vee L_i =\vee {(\overline{\Sigma_i L_i})}$, where $L_i(i\in I)$ is a collection of closed left ideals of $A$.
If for closed left ideals $L_1$ and $L_2$ of $A$, there is some closed left ideals $L$ such that $$(\vee L_1)\cup(\vee L_2)=\vee (L)\tag{1},$$ then $\{\mathcal N\backslash(\vee L)|L \mbox{ is a closed left ideal of }A\}$ forms a topology on $\mathcal N$. And since $$L=\cap \{N_\tau\in \mathcal N|L\subset N_\tau\},$$ we will obtain a bijection between closed left ideals of $A$ and closed set of $\mathcal N$.
Dose such a closed left ideal $L$ in eq(1) exist?
My attempt: Let $L=L_1L_2$, then $\vee(L)\subset \vee L_1\cup\vee L_2$, but the other direction cannot be verified; let $L=(L_1:A)L_2$, where $L_1:A=\{a\in A| aA\subset L_1\}$ is the largeset ideal contained in $L_1$, then $\vee(L)\supset \vee L_1\cup\vee L_2$ but another direction cannot be verified either.