Type III representations and automorphisms of some separable nuclear C∗-algebras

Powers proved decades ago that if two cyclic representations π1 and π2 of a UHF algebra A satisfy that π1(A)″≅π2(A)″=M, there is an automorphism α of A such that π1α and π2 are quasi-equivalent. This was recently extended to the class of simple separable C∗-algebras when M is of type I. In this paper we extend this result to some class of simple separable nuclear C∗-algebras when M is of type III. In particular, we show that the above result obtained by Powers for the UHF algebras also holds for the class of purely infinite simple separable C∗-algebras classified by Kirchberg and Phillips and for the class of approximately homogeneous simple separable unital C∗-algebra with unique tracial state.