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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 α 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 * (,A,α) also has these three properties. We also present examples of C * -algebras A with automorphisms α which satisfy the above assumptions, but such that C * (,A,α) does not have tracial rank zero.

MSC:
46L55Noncommutative dynamical systems
46L80K-theory and operator algebras