A description of Hilbert 𝐶*-modules in which all closed submodules are orthogonally closed

Abstract

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