A scheme for topological phases of the Weyl $C^{\ast }$-algebra

In this work, we introduce a classification scheme for topological phases of matter based on the topology of the space of pure states of a model $C^{\ast }$-algebra. Under it, topological phases are described by homotopy classes of sections of certain fiber bundles of (pure) states. Applying this classification procedure on states of the Weyl $C^{\ast }$-algebra that are invariant under translations by a lattice, we recover the $K$-theoretic classification of gapped spectral projectors for topological insulators of types A and AI, thus essentially generalizing this notion.