Classification of homomorphisms and dynamical systems

Let A be a unital simple C∗-algebra with tracial rank zero and X be a compact metric space. Suppose that h1, h2 : C(X) → A are two unital monomorphisms. We show that h1 and h2 are approximately unitarily equivalent if and only if [h1] = [h2] in KL(C(X), A) and τ ◦ h1(f) = τ ◦ h2(f) for every f ∈ C(X) and every trace τ of A.Adopting a theorem of Tomiyama, we introduce a notion of approximate conjugacy for minimal dynamical systems. Let α, β : X → X be two minimal homeomorphisms. Using the above mentioned result, we show that two dynamical systems are approximately conjugate in that sense if and only if a K-theoretical condition is satisfied. In the case that X is the Cantor set, this notion coincides with strong orbit equivalence of Giordano, Putnam and Skau and the K-theoretical condition is equivalent to saying that the associate crossed product C∗-algebras are isomorphic. Another application of the above mentioned result is given for C∗-dynamical systems related to a problem of Kishimoto. Let A be a unital simple AH-algebra with no dimension growth and with real rank zero, and let α ∈ Aut(A). We prove that if αr fixes a large subgroup of K0(A) and has the tracial Rokhlin property then A ⋊α Z is again a unital simple AH-algebra with no dimension growth and with real rank zero.