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.

我们展示了Raeburn在他的连续迹线的Picard组的计算中引入的希尔伯特模块的粘合构造 $C^{⁎}$-algebra（1981）[14]可应用于任意 $C^{⁎}$-代数，通过Haagerup张量积的代数论证。通过将Hilbert模块上的粘合数据类别识别为$C^{⁎}$-代数超过comodules的代数 $C^{⁎}$-coalgebras，从代数几何中给出标准构造的希尔伯特模块版本。结果表明，如果两个$C^{⁎}$-代数具有相同的原始理想空间T，并且在Morita上等价于T上的2-cocycle ，因此它们相对于T的皮卡德群是同构的。