Asymptotically Unitary Equivalence and Asymptotically Inner Automorphisms

Let C be a unital AH-algebra and let A be a unital separable simple C∗-algebra with tracial rank zero. Suppose that φ1, φ2 : C → A are two unital monomorphisms. We show that there is a continuous path of unitaries {ut : t ∈ [0,∞)} of A such that lim t→∞ u∗tφ1(a)ut = φ2(a) for all a ∈ C if and only if [φ1] = [φ2] in KK(C,A), τ ◦ φ1 = τ ◦ φ2 for all τ ∈ T (A) and the rotation map η̃φ1,φ2 associated with φ1 and φ2 is zero. In particular, an automorphism α on a unital separable simple C∗-algebra A in N with tracial rank zero is asymptotically inner if and only if [α] = [idA] in KK(A,A) and the rotation map η̃φ1,φ2 is zero. Let A be a unital AH-algebra (not necessarily simple) and let α ∈ Aut(A) be an automorphism. As an application, we show that the associated crossed product A ⋊α Z can be embedded into a unital simple AF-algebra if and only if A admits a strictly positive α-invariant tracial state.