Extensions of Hilbert C*-modules: Classification in simple cases

The development of the theory of extensions of Hilbert C*-modules was initiated by Bakic and Guljas. An easy observation shows that, if the underlying C*-algebra extension is commutative and the Hilbert C*-modules are projective of finite type, then the algebraic properties of the corresponding Busby invariant enable one to identify extensions with isometric mappings of the corresponding vector bundles. For the case in which the Hilbert C*-modules are free of rank one, we evaluate the set of extensions in topological terms.