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Tracially AF $C^*$-algebras


Author: Huaxin Lin
Journal: Trans. Amer. Math. Soc. 353 (2001), 693-722
MSC (2000): Primary 46L05, 46L35
DOI: https://doi.org/10.1090/S0002-9947-00-02680-5
Published electronically: September 15, 2000
MathSciNet review: 1804513
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Abstract:

Inspired by a paper of S. Popa and the classification theory of nuclear $C^*$-algebras, we introduce a class of $C^*$-algebras which we call tracially approximately finite dimensional (TAF). A TAF $C^*$-algebra is not an AF-algebra in general, but a ``large'' part of it can be approximated by finite dimensional subalgebras. We show that if a unital simple $C^*$-algebra is TAF then it is quasidiagonal, and has real rank zero, stable rank one and weakly unperforated $K_0$-group. All nuclear simple $C^*$-algebras of real rank zero, stable rank one, with weakly unperforated $K_0$-group classified so far by their $K$-theoretical data are TAF. We provide examples of nonnuclear simple TAF $C^*$-algebras. A sufficient condition for unital nuclear separable quasidiagonal $C^*$-algebras to be TAF is also given. The main results include a characterization of simple rational AF-algebras. We show that a separable nuclear simple TAF $C^*$-algebra $A$ satisfying the Universal Coefficient Theorem and having $K_1(A)=0$ and $K_0(A)=\mathbf{Q}$ is isomorphic to a simple AF-algebra with the same $K$-theory.


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Additional Information

Huaxin Lin
Affiliation: Department of Mathematics, East China Normal University, Shanghai, China
Address at time of publication: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222
Email: hxlin@noether.uoregon.edu

DOI: https://doi.org/10.1090/S0002-9947-00-02680-5
Keywords: TAF $C^*$-algebras, real rank zero, simple
Received by editor(s): May 5, 1998
Received by editor(s) in revised form: April 3, 1999
Published electronically: September 15, 2000
Additional Notes: Research partially supported by NSF grants DMS 9801482.
Article copyright: © Copyright 2000 American Mathematical Society

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