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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Classification of homomorphisms and dynamical systems


Author: Huaxin Lin
Journal: Trans. Amer. Math. Soc. 359 (2007), 859-895
MSC (2000): Primary 46L35, 46L55
Published electronically: September 12, 2006
MathSciNet review: 2255199
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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

$\displaystyle [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.


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Additional Information

Huaxin Lin
Affiliation: Department of Mathematics, East China Normal University, Shanghai, People’s Republic of China
Address at time of publication: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222

DOI: http://dx.doi.org/10.1090/S0002-9947-06-03932-8
PII: S 0002-9947(06)03932-8
Received by editor(s): April 22, 2004
Received by editor(s) in revised form: January 6, 2005
Published electronically: September 12, 2006
Dedicated: Dedicated to George Elliott on his 60th birthday
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.