We associate to each unital C∗-algebra A a geometric object---a diagram
of topological spaces representing quotient spaces of the noncommutative space
underlying A---meant to serve the role of a generalized Gel'fand spectrum.
After showing that any functor F from compact Hausdorff spaces to a suitable
target category can be applied directly to these geometric objects to
automatically yield an extension F~ which acts on all unital
C∗-algebras, we compare a novel formulation of the operator K0​ functor to
the extension K~ of the topological K-functor.Comment: 14 page