Ergodic Theory and Dynamical Systems

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

a1 Department of Mathematical Sciences, Ritsumeikan University, Kusatsu, Shiga, 525-8577, Japan (e-mail:
a2 Department of Mathematics, University of Oregon, Eugene, OR 97403-1222, USA (e-mail:

Article author query
osaka h   [Google Scholar] 
phillips nc   [Google Scholar] 


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. Let $A$ be a stably finite simple unital $C^*$-algebra, and let $\alpha$ be an automorphism of $A$ which has the tracial Rokhlin property. Suppose $A$ has real rank zero and stable rank one, and suppose that the order on projections over $A$ is determined by traces. Then the crossed product algebra $C^* (\mathbb{Z}, A, \alpha)$ also has these three properties. We also present examples of $C^*$-algebras $A$ with automorphisms $\alpha$ which satisfy the above assumptions, but such that $C^* (\mathbb{Z}, A, \alpha)$ does not have tracial rank zero.

(Published Online September 11 2006)
(Received April 15 2005)
(Revised January 13 2006)