Ergodic Theory and Dynamical Systems



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


HIROYUKI OSAKA a1 and N. CHRISTOPHER PHILLIPS a2
a1 Department of Mathematical Sciences, Ritsumeikan University, Kusatsu, Shiga, 525-8577, Japan (e-mail: osaka@se.ritsumei.ac.jp)
a2 Department of Mathematics, University of Oregon, Eugene, OR 97403-1222, USA (e-mail: ncp@darkwing.uoregon.edu)

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

Abstract

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)