  We consider C*-algebras associated with stable and unstable equivalence in
hyperbolic dynamical systems known as Smale spaces. These systems include
shifts of finite type, in which case these C*-algebras are both AF-algebras.
These algebras have fundamental representations on a single Hilbert space
(subject to a choice of periodic points) which have a number of special
properties. In particular, the product between any element of the first algebra
with one from the second is compact. In addition, there is a single unitary
operator which implements actions on both. Here, under the hypothesis that the
system is mixing, we show that the (semi-finite) traces on these algebras may
be obtained through a limiting process and the usual operator trace.
