Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-06-03932-8
Published electronically: September 12, 2006
MathSciNet review: 2255199
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $f\in C(X)$ 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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 46L35, 46L55

Retrieve articles in all journals with MSC (2000): 46L35, 46L55


Additional Information

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

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.