  The notion of a topological phase of an insulator is based on the concept of
homotopy between Hamiltonians. It therefore depends on the choice of a
topological space to which the Hamiltonians belong. We advocate that this space
should be the $C^*$-algebra of observables. We relate the symmetries of
insulators to graded real structures on the observable algebra and classify the
topological phases using van Daele's formulation of $K$-theory. This is related
but not identical to Thiang's recent approach to classify topological phases by
$K$-groups in Karoubi's formulation.
