Let \(\mathscr {E}\) be a Hilbert \(C^*\) -module over a \(C^*\) -algebra \(\mathscr {A}\) . For each positive linear functional \(\omega \) on \(\mathscr {A}\) , we consider the localization of \(\mathscr {E}\) , denoted by \(\mathscr {E}_\omega \) , 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 an orthogonally complemented submodule, and let \(\omega = \sum _{j=1}^{\infty } \lambda _j \omega _j,\) where \(\lambda _j > 0\) , \(\sum _{j=1}^{\infty } \lambda _j = 1\) , and each \(\omega _j\) is a positive linear functional 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 \(W^*\) -module \(\mathscr {E}\) over a \(W^*\) -algebra \(\mathscr {A}\) . Let \(\iota _\omega : \mathscr {E} \rightarrow \mathscr {E}_\omega \) be the natural quotient map. We consider the following separation problem: “Does there exist a vector state \(\omega \) such that \(\iota _\omega (\mathscr {L})\) is not dense in \(\mathscr {E}_\omega \) ?” Among other results, we prove that if \(\mathscr {E}\) is a self-dual Hilbert \(W^*\) -module and \(\mathscr {E} \setminus \mathscr {L}\) has nonempty \(\text {weak}^*\) -interior, then there exist vector states \(\omega _1,\ldots ,\omega _n\) such that \(\iota _\omega (\mathscr {L}_1)\) is not dense in the unit ball of \(\mathscr {E}_\omega \) for every \(\omega \in \textrm{co}\{\omega _1,\ldots ,\omega _n\}\) with strictly positive coefficients, where \(\mathscr {L}_1\) denotes the unit ball of \(\mathscr {L}\) .
