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The tracial Rokhlin property for actions of finite groups on C*-algebras
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 133, Number 3, June 2011
- pp. 581-636
- 10.1353/ajm.2011.0016
- Article
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We define ``tracial'' analogs of
the Rokhlin property for actions of finite groups,
approximate representability of actions of finite abelian groups,
and approximate innerness.
We prove the following four
analogs of related ``nontracial'' results. $\bullet$
The crossed product of an infinite dimensional
simple separable unital C*-algebra with tracial rank zero
by an action of a finite group with the tracial Rokhlin property
again has tracial rank zero. $\bullet$
An outer action of a finite abelian group on
an infinite dimensional
simple separable unital C*-algebra has the tracial Rokhlin property
if and only if its dual is tracially approximately representable,
and is tracially approximately representable
if and only if its dual has the tracial Rokhlin property. $\bullet$
If a strongly tracially approximately inner action of a
finite cyclic group on an infinite dimensional
simple separable unital C*-algebra has the tracial Rokhlin property,
then it is tracially approximately representable. $\bullet$
An automorphism of an infinite dimensional
simple separable unital C*-algebra $A$ with tracial rank zero
is tracially approximately inner
if and only if it is the identity on $K_0 (A)$ mod infinitesimals.