A description of Hilbert $C^\ast$-modules in which all closed submodules are orthogonally closed

Abstract:

Let $A$, $B$ be $C^*$-algebras and $E$ a full Hilbert $A$-$B$-bimodule such that every closed right submodule $E_{0}\subseteq E$ is orthogonally closed, i.e., $E_{0}=(E_{0}^{\perp })^{\perp }$. Then there are families of Hilbert spaces $\{\mathcal {H}_{i}\}$, $\{\mathcal {V}_{i}\}$ such that $A$ and $B$ are isomorphic to $c_{0}$-direct sums $\sum \mathcal {K}(\mathcal {V}_{i})$, resp. $\sum \mathcal {K}(\mathcal {H}_{i})$, and $E$ is isomorphic to the outer direct sum $\sum _{0}\mathcal {K}(\mathcal {H}_{i},\mathcal {V}_{i})$.