A separation theorem for Hilbert $W^*$-modules

Let $\mathscr E$ be a Hilbert $\mathscr A$-module over a $C^*$-algebra $\mathscr A$. For each positive linear functional $\omega$ on $\mathscr A$, we consider the localization $\mathscr E_\omega$ of $\mathscr E$, which is the completion of the quotient space $\mathscr E/\mathscr {N}_\omega$, where $\mathscr N_\omega=\{x\in \mathscr E:\omega\langle x,x\rangle=0\}$. Let $\mathscr H$ and $\mathscr K$ be closed submodules of $\mathscr E$ such that $\mathscr H\cap \mathscr K$ is orthogonally complemented, and let $\omega=\sum_{j=1}^{\infty}\lambda_j\omega_j$, where $\lambda_j>0$, $\sum_{j=1}^{\infty}\lambda_j=1$, and $\omega_j$'s are positive linear functionals on $\mathscr A$. We prove that if $(\mathscr H\cap \mathscr K)_{\omega_j}=\mathscr H_{\omega_j}\cap \mathscr K_{\omega_j}$ for each $j$, then $$(\mathscr H\cap \mathscr K)_\omega=\mathscr H_\omega\cap \mathscr K_\omega\,.$$ Furthermore, let $\mathscr L$ be a closed submodule of a Hilbert $\mathscr A$-module $\mathscr E$ over a $W^*$-algebra $\mathscr A$. We pose the following separation problem: ``Does there exist a normal state $\omega$ such that $\iota_\omega (\mathscr L)$ is not dense in $\mathscr E_\omega$?'' In this paper, among other results, we give an affirmative answer to this problem, when $\mathscr E$ is a self-dual Hilbert $C^*$-module over a $W^*$-algebra $\mathscr A$ such that $\mathscr E\backslash \mathscr L$ has a nonempty interior with respect to the weak$^*$-topology. This is a step toward answering the above problem.