Tracially AF 𝐶*-algebras

Abstract

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.