From Topology to Noncommutative Geometry: $K$-theory

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 $\tilde{F}$ which acts on all unital $C^*$-algebras, we compare a novel formulation of the operator $K_0$ functor to the extension $\tilde K$ of the topological $K$-functor.