Gluing Hilbert C*-modules over the primitive ideal space

TL;DR

This paper extends the gluing construction for Hilbert modules to all C*-algebras using algebraic methods, linking it to descent theory and showing implications for Picard groups under Morita equivalence.

Contribution

It generalizes the gluing construction for Hilbert modules to arbitrary C*-algebras and connects it with descent theory and C*-coalgebras.

Findings

Gluing construction applies to all C*-algebras via algebraic argument.

Categories of gluing data are equivalent to categories of comodules over C*-coalgebras.

Picard groups are invariant under Morita equivalence up to a 2-cocycle on the primitive ideal space.

Abstract

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 (Trans. Amer. Math. Soc., 1981) 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.