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Zbl 1067.46062
Lin, Huaxin; Osaka, Hiroyuki
The Rokhlin property and the tracial topological rank. (English)
[J] J. Funct. Anal. 218, No. 2, 475-494 (2005). ISSN 0022-1236

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} $\text{TR}(A)$. The {tracially cyclic Rokhlin property} is defined for an automorphism $\alpha$ of $A$. It is shown that if $\alpha$ satisfies the tracially cyclic Rokhlin property and $\text{TR}(A)\leq 1$ then $\text{TR}(A\rtimes_\alpha\Bbb Z)\leq 1$. It is also shown that if $A$ has a unique tracial state and $\alpha^m$ is uniformly outer for each $m\neq 0$ and $\alpha^r$ is approximately inner for some $r>0$ then $\alpha$ satisfies the tracial cyclic Rokhlin property. This result is applied to prove the following conjecture of Kishimoto: if $A$ is a unital simple $A\Bbb T$-algebra of real rank zero and $\alpha$ is approximately inner and satisfies a version of the Rokhlin property, then $A\rtimes_\alpha\Bbb Z$ is again an $A\Bbb T$-algebra of real rank zero.
[V. M. Manuilov (Moskva)]

MSC 2000:: *46L55 Noncommutative dynamical systems
46L35 Classifications and factors of C*-algebras

Keywords: simple $C^*$-algebra; Rohlin property; tracial topological rank

Citations: Zbl 1015.46031

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