Classification of homomorphisms and dynamical systems

Let $A$ be a unital simple $C^*$-algebra, with tracial rank zero and let $X$ be a compact metric space. Suppose that $h_1, h_2: C(X)\to A$ are two unital monomorphisms. We show that $h_1$ and $h_2$ are approximately unitarily equivalent if and only if

$[h_1]=[h_2] \,\,\,{\rm in}\,\,\, KL(C(X),A)\,\,\,\,\,\, {\rm and} \,\,\,\,\,\, \tau\circ h_1(f)=\tau\circ h_2(f)$

for every $f\in C(X)$ and every trace $\tau$ of $A.$ Inspired by a theorem of Tomiyama, we introduce a notion of approximate conjugacy for minimal dynamical systems. Let $X$ be a compact metric space and let $\alpha, \beta: X\to X$ 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 the 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 $\alpha\in Aut(A).$ We prove that if $\alpha^r$ fixes a large subgroup of $K_0(A)$ and has the tracial Rokhlin property, then $A\rtimes_{\alpha}\mathbb{Z}$ is again a unital simple AH-algebra with no dimension growth and with real rank zero.