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• W2170208706 abstract "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] textrm {in} KL(C(X),A) textrm {and} tau circ h_1(f)=tau circ h_2(f) ] for every $fin C(X)$ 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 $alpha , beta : Xto X$ 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 $Artimes _{alpha }mathbb {Z}$ is again a unital simple AH-algebra with no dimension growth and with real rank zero." @default.
• W2170208706 created "2016-06-24" @default.
• W2170208706 creator A5078227996 @default.
• W2170208706 date "2006-09-12" @default.
• W2170208706 modified "2023-09-25" @default.
• W2170208706 title "Classification of homomorphisms and dynamical systems" @default.
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• W2170208706 doi "https://doi.org/10.1090/s0002-9947-06-03932-8" @default.
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