K-Theory for Topological Solid State Systems

Abstract

The first part of the chapter reviews the K-theory of unital and non-unital C$^*$-algebras, particularly, the K-groups and their standard characterization, the six-term exact sequences and their connecting maps as well as the suspensions and Bott periodicity. In the second part, the analysis is specialized to the observable algebras defined in Chap. 3. Using the Pimsner-Voiculescu sequence, this allows to present the generators of the K-groups in detail. In the third part, various connecting maps for solid state systems are computed explicitly.