We study the hyperbolic dynamical systems known as Smale spaces. More specifically we investigate the C*-algebras constructed from these systems. The K group of one of these algebras has a natural ring structure arising from an asymptotically abelian property. The K groups of the other algebras are then modules over this ring. In the case of a shift of finite type we compute these structures explicitly and show that the stable and unstable algebras exhibit a certain type of duality as modules. We also investigate the Bowen measure and its stable and unstable components with respect to resolving factor maps, and prove several results about the traces that arise as integration against these measures. Specifically we show that the trace is a ring/module homomorphism into R and prove a result relating these integration traces to an asymptotic of the usual trace of an operator on a Hilbert space.