Stable and real rank for crossed products by automorphisms with the tracial Rokhlin property

We introduce the tracial Rokhlin property for automorphisms of stably finite simple unital C*-algebras containing enough projections. This property is formally weaker than the various Rokhlin properties considered by Herman and Ocneanu, Kishimoto, and Izumi. Our main results are as follows. Consider a stably finite simple unital C*-algebra, and an automorphism of it which has the tracial Rokhlin property. Suppose the algebra has real rank zero and stable rank one, and suppose that the order on projections over the algebra is determined by traces. Then the crossed product by the automorphism also has these three properties. We also present examples of C*-algebras and automorphisms which satisfy the above assumptions, but such that crossed product algebras do not have tracial rank zero.