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Approximate unitary equivalence in simple $ C^*$-algebras of tracial rank one


Author: Huaxin Lin
Journal: Trans. Amer. Math. Soc. 364 (2012), 2021-2086
MSC (2010): Primary 46L35
DOI: https://doi.org/10.1090/S0002-9947-2011-05431-0
Published electronically: December 2, 2011
MathSciNet review: 2869198
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ C$ be a unital AH-algebra and let $ A$ be a unital separable simple $ C^*$-algebra with tracial rank no more than one. Suppose that $ \phi , \psi : C\to A$ are two unital monomorphisms. With some restriction on $ C,$ we show that $ \phi $ and $ \psi $ are approximately unitarily equivalent if and only if

$\displaystyle [\phi ]$ $\displaystyle =$ $\displaystyle [\psi ]\,\,\,\textup {in}\,\,\, KL(C,A),$  
$\displaystyle \tau \circ \phi$ $\displaystyle =$ $\displaystyle \tau \circ \psi \ \textup {for all tracial states of}\,\,\, A\ $$\displaystyle \text {and}$  
$\displaystyle \phi ^{\ddag }$ $\displaystyle =$ $\displaystyle \psi ^{\ddag },$  

where $ \phi ^{\ddag }$ and $ \psi ^{\ddag }$ are homomorphisms from $ U(C)/CU(C)\to U(A)/CU(A)$ induced by $ \phi $ and $ \psi ,$ respectively, and where $ CU(C)$ and $ CU(A)$ are closures of the subgroup generated by commutators of the unitary groups of $ C$ and $ B.$

A more practical but approximate version of the above is also presented.


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Additional Information

Huaxin Lin
Affiliation: Department of Mathematics, East China Normal University, Shanghai, People’s Republic of China – and – Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Email: hlin@uoregon.edu

DOI: https://doi.org/10.1090/S0002-9947-2011-05431-0
Received by editor(s): February 12, 2008
Received by editor(s) in revised form: June 3, 2010
Published electronically: December 2, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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