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Stable and real rank for crossed products by automorphisms with the tracial Rokhlin property. (English) Zbl 1116.46059
Summary: 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 that $A$ 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.

46L55Noncommutative dynamical systems
46L80$K$-theory and operator algebras
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