# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $\text{TR}\left(A\right)$. 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 $\text{TR}\left(A\right)\le 1$ then $\text{TR}\left(A{⋊}_{\alpha }ℤ\right)\le 1$. It is also shown that if $A$ has a unique tracial state and ${\alpha }^{m}$ is uniformly outer for each $m\ne 0$ 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 $A𝕋$-algebra of real rank zero and $\alpha$ is approximately inner and satisfies a version of the Rokhlin property, then $A{⋊}_{\alpha }ℤ$ is again an $A𝕋$-algebra of real rank zero.
##### MSC:
 46L55 Noncommutative dynamical systems 46L35 Classifications of ${C}^{*}$-algebras