Gluing Hilbert C⁎-modules over the primitive ideal space

We show that the gluing construction for Hilbert modules introduced by Raeburn in his computation of the Picard group of a continuous-trace C⁎-algebra (1981) [14] can be applied to arbitrary C⁎-algebras, via an algebraic argument with the Haagerup tensor product. We put this result into the context of descent theory by identifying categories of gluing data for Hilbert modules over C⁎-algebras with categories of comodules over C⁎-coalgebras, giving a Hilbert-module version of a standard construction from algebraic geometry. As a consequence we show that if two C⁎-algebras have the same primitive ideal space T, and are Morita equivalent up to a 2-cocycle on T, then their Picard groups relative to T are isomorphic.