Spectral triples for AF C*-algebras and metrics on the cantor set

An AF C -algebra has a natural filtration as an increasing se- quence of finite dimensional C -algebras. We show that it is possible to con- struct a Dirac operator which relates to this filtration in a natural way and which will induce a metric for the weak*-topology on the state space of the algebra. It turns out that for AF C -algebras, there is no limit to the growth of the eigenvalues of such a Dirac operator. We have obtained a kind of an in- verse to this result, by showing that a phenomenon like this can only occur for AF C -algebras. The results are then applied to a study of the classical Cantor set.