Certain Aperiodic Automorphisms of Unital Simple Projectionless C * -algebras

Let G be an inductive limit of finite cyclic groups and let A be a unital simple projectionless C∗-algebra with K1(A) = G and with a unique tracial state, as constructed based on dimension drop algebras by Jiang and Su. First, we show that any two aperiodic elements in Aut(A)/WInn(A) are conjugate, where WInn(A) means the subgroup of Aut(A) consisting of automorphisms which are inner in the tracial representation. In the second part of this paper, we consider a class of unital simple C∗-algebras with a unique tracial state which contains the class of unital simple AT-algebras of real rank zero with a unique tracial state. This class is closed under inductive limits and under crossed products by actions of Z with the Rohlin property. Let A be a TAF-algebra in this class. We show that for any automorphism α of A there exists an automorphism e α of A with the Rohlin property such that e α and α are asymptotically unitarily equivalent. In its proof we use an aperiodic automorphism of the Jiang-Su algebra.