  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.
