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The Rokhlin property and the tracial topological rank. (English) Zbl 1067.46062
For a unital separable simple C * -algebra A, the first named author has defined in [Proc. London Math. Soc., III. Ser. 83, 199–234 (2001; Zbl 1015.46031)] the tracial topological rank TR(A). The tracially cyclic Rokhlin property is defined for an automorphism α of A. It is shown that if α satisfies the tracially cyclic Rokhlin property and TR(A)1 then TR(A α )1. It is also shown that if A has a unique tracial state and α m is uniformly outer for each m0 and α r is approximately inner for some r>0 then α satisfies the tracial cyclic Rokhlin property. This result is applied to prove the following conjecture of Kishimoto: if A is a unital simple A𝕋-algebra of real rank zero and α is approximately inner and satisfies a version of the Rokhlin property, then A α is again an A𝕋-algebra of real rank zero.
46L55Noncommutative dynamical systems
46L35Classifications of C * -algebras