Inspired by a paper of S. Popa and the classification theory of nuclear
C
∗
C^*
-algebras, we introduce a class of
C
∗
C^*
-algebras which we call tracially approximately finite dimensional (TAF). A TAF
C
∗
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
∗
C^*
-algebra is TAF then it is quasidiagonal, and has real rank zero, stable rank one and weakly unperforated
K
0
K_0
-group. All nuclear simple
C
∗
C^*
-algebras of real rank zero, stable rank one, with weakly unperforated
K
0
K_0
-group classified so far by their
K
K
-theoretical data are TAF. We provide examples of nonnuclear simple TAF
C
∗
C^*
-algebras. A sufficient condition for unital nuclear separable quasidiagonal
C
∗
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
∗
C^*
-algebra
A
A
satisfying the Universal Coefficient Theorem and having
K
1
(
A
)
=
0
K_1(A)=0
and
K
0
(
A
)
=
Q
K_0(A)=\mathbf {Q}
is isomorphic to a simple AF-algebra with the same
K
K
-theory.