The Rokhlin property and the tracial topological rank

Abstract

Let $A$ be a unital separable simple \CA with $\tr(A)\le 1$ and $\alpha$ be an automorphism. We show that if $\alpha$ satisfies the tracially cyclic Rokhlin property then $\tr(A\rtimes_{\alpha}\Z)\le 1.$ We also show that whenever $A$ has a unique tracial state and $\alpha^m$ is uniformly outer for each $m (\not= 0)$ and $\alpha^r$ is approximately inner for some $r>0,$ $\alpha$ satisfies the tracial cyclic Rokhlin property. By applying the classification theory of nuclear \CA s, we use the above result to prove a conjecture of Kishimoto: if $A$ is a unital simple $A{\mathbb T}$-algebra of real rank zero and $\alpha\in \Aut(A)$ which is approximately inner and if $\alpha$ satisfies some Rokhlin property, then the crossed product $A\rtimes_{\alpha}\Z$ is again an $A{\mathbb T}$ -algebra of real rank zero. As a by-product, we find that one can construct a large class of simple \CA s with tracial rank one (and zero) from crossed products.