### Abstract

In order to study continuous models of disordered topological phases, we construct an unbounded Kasparov module and a semifinite spectral triple for the crossed product of a separable C^{∗}-algebra by a twisted ℝ^{d}-action. The spectral triple allows us to employ the non-unital local index formula to obtain the higher Chern numbers in the continuous setting with complex observable algebra. In the case of the crossed product of a compact disorder space, the pairing can be extended to a larger algebra closely related to dynamical localisation, as in the tight-binding approximation. The Kasparov module allows us to exploit the Wiener–Hopf extension and the Kasparov product to obtain a bulk-boundary correspondence for continuous models of disordered topological phases.

Original language | English |
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Article number | 16 |

Journal | Mathematical Physics Analysis and Geometry |

Volume | 21 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2018 Sep 1 |

### Keywords

- Crossed product
- Kasparov theory
- Topological states of matter

### ASJC Scopus subject areas

- Mathematical Physics
- Geometry and Topology