Classification of Simple Tracially AF $C^*$-Algebras

http://dx.doi.org/10.4153/CJM-2001-007-8
Canad. J. Math. 53(2001), 161-194

  Published:2001-02-01
 Printed: Feb 2001

We prove that pre-classifiable (see 3.1) simple nuclear tracially AF \CA s (TAF) are classified by their $K$-theory. As a consequence all simple, locally AH and TAF \CA s are in fact AH algebras (it is known that there are locally AH algebras that are not AH). We also prove the following Rationalization Theorem. Let $A$ and $B$ be two unital separable nuclear simple TAF \CA s with unique normalized traces satisfying the Universal Coefficient Theorem. If $A$ and $B$ have the same (ordered and scaled) $K$-theory and $K_0 (A)_+$ is locally finitely generated, then $A \otimes Q \cong B \otimes Q$, where $Q$ is the UHF-algebra with the rational $K_0$. Classification results (with restriction on $K_0$-theory) for the above \CA s are also obtained. For example, we show that, if $A$ and $B$ are unital nuclear separable simple TAF \CA s with the unique normalized trace satisfying the UCT and with $K_1(A) = K_1(B)$, and $A$ and $B$ have the same rational (scaled ordered) $K_0$, then $A \cong B$. Similar results are also obtained for some cases in which $K_0$ is non-divisible such as $K_0(A) = \mathbf{Z} [1/2]$.