Spectral Theory for Non-full Commutative C*-categories

We extend the spectral theory of commutative C*-categories to the non full-case, introducing a suitable notion of spectral spaceoid provinding a duality between a category of “non-trivial” -functors of non-full commutative C-categories and a category of Takahashi morphisms of “non-full spaceoids” (here defined). As a byproduct we obtain a spectral theorem for a non-full generalization of imprimitivity Hilbert C*-bimodules over commutative unital C*-algebras via continuous sections vanishing at infinity of a Hilbert C*-line-bundle over the graph of a homeomorphism between open subsets of the corresponding Gel’fand spectra of the C*-algebras.