Mathematics – Operator Algebras
Scientific paper
2004-04-01
Mathematics
Operator Algebras
Scientific paper
Let $A$ be a unital simple C*-algebra with tracial rank zero and $X$ 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 $A.$ Adopting a theorem of Tomiyama, we introduce a notion of approximate conjugacy for minimal dynamical systems. Let $X$ be a compact metric space and $\alpha, \beta: X\to X$ 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 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 $A\rtimes_{\alpha}\Z$ is again a unital simple AH-algebra with no dimension growth and with real rank zero.
No associations
LandOfFree
Classification of homomorphisms and dynamical systems does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Classification of homomorphisms and dynamical systems, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Classification of homomorphisms and dynamical systems will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-145269