In this work, we provide a classification scheme for topological phases of certain systems whose observable algebra is described by a trivial \(C^*\) -bundles. The classification is based on the study of the homotopy classes of configurations, which are maps from a quantum parameter space to the space of pure states of a reference fiber \(C^*\) -algebra. Both the quantum parameter space and the fiber algebra are naturally associated with the observable algebra. A list of various examples described in the last section shows that the common classification scheme of non-interacting topological insulators of type A is recovered inside this new formalism.
