Gluing Hilbert C*-modules over the primitive ideal space

Tyrone Crisp

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arXiv · 2006.04615

Gluing Hilbert C*-modules over the primitive ideal space

Tyrone Crisp

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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.

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Tyrone Crisp. 2020-06-08. Gluing Hilbert C*-modules over the primitive ideal space. https://arxiv.org/abs/2006.04615

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