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.
We show that every separable nuclear residually finite dimensional -algebras satisfying the Universal Coefficient Theorem can be embedded into a unital separable simple AF-algebra.
