Spectral Theory for Non-full Commutative C*-categories

TL;DR

This paper generalizes spectral theory to non-full commutative C*-categories, establishing a duality with spaceoids and deriving a spectral theorem for imprimitivity bimodules over commutative C*-algebras.

Contribution

It introduces a spectral spaceoid concept for non-full C*-categories and establishes a duality with non-trivial *-functors, extending spectral theory.

Findings

Established a duality between non-full commutative C*-categories and spaceoids.

Derived a spectral theorem for non-full imprimitivity Hilbert C*-bimodules.

Connected spectral theory with continuous sections of Hilbert C*-line-bundles.

Abstract

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.