The tracial Rokhlin property for actions of finite groups on C*-algebras

Abstract

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.