A separation theorem for Hilbert $W^{\ast }$-modules

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