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

Abstract

Let $\mathscr E$ be a Hilbert $\mathscr A$-module over a $C^*$-algebra $\mathscr A$. For each positive linear functional $ω$ on $\mathscr A$, we consider the localization $\mathscr E_ω$ of $\mathscr E$, which is the completion of the quotient space $\mathscr E/\mathscr {N}_ω$, where $\mathscr N_ω=\{x\in \mathscr E:ω\langle x,x\rangle=0\}$. Let $\mathscr H$ and $\mathscr K$ be closed submodules of $\mathscr E$ such that $\mathscr H\cap \mathscr K$ is orthogonally complemented, and let $ω=\sum_{j=1}^{\infty}λ_jω_j$, where $λ_j>0$, $\sum_{j=1}^{\infty}λ_j=1$, and $ω_j$'s are positive linear functionals on $\mathscr A$. We prove that if $(\mathscr H\cap \mathscr K)_{ω_j}=\mathscr H_{ω_j}\cap \mathscr K_{ω_j}$ for each $j$, then \[ (\mathscr H\cap \mathscr K)_ω=\mathscr H_ω\cap \mathscr K_ω\,. \] 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 $ω$ such that $ι_ω(\mathscr L)$ is not dense in $\mathscr E_ω$?'' 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.