The Rokhlin property and the tracial topological rank

Let A be a unital separable simple C*-algebra with TR(A) less than or equal to 1 and alpha be an automorphism. We show that if a satisfies the tracially cyclic Rokhlin property then TR(A !!(alpha)Z) less than or equal to 1. We also show that whenever A has a unique tracial state and alpha(m) is uniformly outer for each m(not equal0) and alpha(r) is approximately inner for some r>0, a satisfies the tracial cyclic Rokhlin property. By applying the classification theory of nuclear C*-algebras, we use the above result to prove a conjecture of Kishimoto: if A is a unital simple AT-algebra of real rank zero and alphais an element ofAut(A) which is approximately inner and if a satisfies some Rokhlin property, then the crossed product A !!(alpha)Z is again an AT-algebra of real rank zero. As a by-product, we find that one can construct a large class of simple C*-algebras with tracial rank one (and zero) from crossed products. (C) 2004 Elsevier Inc. All rights reserved.