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Zbl 1116.46059
Osaka, Hiroyuki; Phillips, N. Christopher
Stable and real rank for crossed products by automorphisms with the tracial Rokhlin property. (English)
[J] Ergodic Theory Dyn. Syst. 26, No. 5, 1579-1621 (2006). ISSN 0143-3857; ISSN 1469-4417/e

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.

MSC 2000:: *46L55 Noncommutative dynamical systems
46L80 K-theory and operator algebras

Keywords: tracial Rohlin property; real rank; stable rank

Citations: Zbl 0578.46058; Zbl 1021.46049; Zbl 0824.46070; Zbl 0902.46031

Cited in: Zbl 1119.46050

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