Asymptotically unitary equivalence and asymptotically inner automorphisms

2007/03/20 by Lin, Huaxin
#46L05 #46L35 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.math/0703610

Abstract

Let C be a unital AH-algebra and let A be a unital separable simple \CA with tracial rank zero. Suppose that ϕ₁, ϕ₂: C→ A are two unital monomorphisms. We show that there is a continuous path of unitaries \u_t: t∈ [0, ∞)\ of A such that lim_t→∞u_t^*ϕ₁(a)u_t=ϕ₂(a)\tforal 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 η_ϕ1,ϕ2 associated with ϕ₁ and ϕ₂ is zero. In particular, an automorphism \af on a unital separable simple \CA A in \cal N with tracial rank zero is asymptotically inner if and only if [\af]=[\rm id_A] in KK(A,A) and the rotation map η_{ϕ1, ϕ2} is zero. Let A be a unital AH-algebra (not necessarily simple) and let \af∈ Aut(A) be an automorphism. As an application, we show that the associated crossed product A\rtimes_\af\Z can be embedded into a unital simple AF-algebra if and only if A admits a strictly positive \af-invariant tracial state.

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