Crossed Products by Finite Cyclic Group Actions with the Tracial Rokhlin Property

We define the tracial Rokhlin property for actions of finite cyclic groups on stably finite simple unital C*-algebras. We prove that the crossed product of a stably finite simple unital C*-algebra with tracial rank zero by an action with this property again has tracial rank zero. Under a kind of weak approximate innerness assumption and one other technical condition, we prove that if the action has the tracial Rokhlin property and the crossed product has tracial rank zero, then the original algebra has tracial rank zero. We give examples showing how the tracial Rokhlin property differs from the Rokhlin property of Izumi. We use these results, together with work of Elliott-Evans and Kishimoto, to give an inductive proof that every simple higher dimensional noncommutative torus is an AT algebra. We further prove that the crossed product of every simple higher dimensional noncommutative torus by the flip is an AF algebra, and that the crossed products of irrational rotation algebras by the standard actions of Z3, Z4, and Z6 are simple AH algebras with real rank zero. In the case of Z4, we recover Walters’ result that the crossed product is AF for a dense Gδ-set of rotation numbers.