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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.

MSC:

46L55 Noncommutative dynamical systems
46L80 \(K\)-theory and operator algebras (including cyclic theory)
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