Tracially AF -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 | References | Similar Articles | Additional Information

Inspired by a paper of S. Popa and the classification theory of nuclear -algebras, we introduce a class of -algebras which we call tracially approximately finite dimensional (TAF). A TAF -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 -algebra is TAF then it is quasidiagonal, and has real rank zero, stable rank one and weakly unperforated -group. All nuclear simple -algebras of real rank zero, stable rank one, with weakly unperforated -group classified so far by their -theoretical data are TAF. We provide examples of nonnuclear simple TAF -algebras. A sufficient condition for unital nuclear separable quasidiagonal -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 -algebra satisfying the Universal Coefficient Theorem and having and is isomorphic to a simple AF-algebra with the same -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