DISCRETE NON-COMMUTATIVE GEL’FAND-NA˘IMARK DUALITY

Abstract

We present, in a simplified setting, a non-commutative version of the wellknown Gel’fand-Na˘ımark duality (between the categories of compact Hausdorff
topological spaces and commutative unital C*-algebras), where “geometric spectra” consist of suitable finite bundles of one-dimensional C*-categories equipped
with a transition amplitude structure satisfying saturation conditions. Although this discrete duality actually describes the trivial case of finite-dimensional C*-algebras, the structures are here developed at a level of generality adequate for theformulation of a general topological/uniform Gel’fand-Na˘ımark duality, fully addressed in a companion work.