Comparison between two approaches to classify topological insulators using K-theory

We compare two approaches which use K-theory for C*-algebras to classify symmetry protected topological phases of quantum systems described in the one particle approximation. In the approach by Kellendonk, which is more abstract and more general, the algebra remains unspecified and the symmetries are defined using gradings and real structures. In the approach by Alldridge et al., the algebra is physically motivated and the symmetries implemented by generators which commute with the Hamiltonian. Both approaches use van Daele’s version of K-theory. We show that the second approach is a special case of the first one. We highlight the role played by two of the symmetries: charge conservation and spin rotation symmetry.