A separation theorem for Hilbert W^*-modules

Let L be a closed submodule of a Hilbert W^*-module E over a C^*-algebra A. We pose a separation problem: Does there exist a normal state ω such that ι_ω ( L) is not dense in E_ω? In this note, among other results, we give an affirmative answer to this problem, when E is a self-dual Hilbert W^*-module such that E L has a nonempty interior with respect to the weak^*-topology.