Tracially AF $C^*$-algebras

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.