Spectral Theory for Non-full Commutative C*-categories
Paolo Bertozzini, Roberto Conti, Wicharn Lewkeeratiyutkul, Kasemsun Rutamorn
TL;DR
This work extends Gel'fand–Naïmark duality to the realm of non-full commutative C*-categories by introducing non-full spaceoids and Takahashi-type morphisms. Central to the framework are the functors Γ (sections) and Σ (spectrum), which yield a contravariant duality between the category A of small commutative non-full C*-categories and the category T of non-full spaceoids, augmented by Gel'fand and evaluation natural isomorphisms. The approach preserves the spectral-theoretic intuition of the full case while accommodating off-diagonal, non-full morphisms, and it delivers a spectral theorem for non-full imprimitivity Hilbert C*-bimodules via the linking C*-category construction. The results provide a robust, categorified geometric viewpoint on non-full spectral data with potential implications for Morita theory, K-theory, and higher-categorical generalizations.
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.