An approximate universal coefficient theorem

Abstract

An approximate Universal Coefficient Theorem (AUCT) for certain C ∗ C^* -algebras is established. We present a proof that Kirchberg-Phillips’s classification theorem for separable nuclear purely infinite simple C ∗ C^* -algebras is valid for C ∗ C^* -algebras satisfying the AUCT instead of the UCT. It is proved that two versions of AUCT are in fact the same. We also show that C ∗ C^* -algebras that are locally approximated by C ∗ C^* -algebras satisfying the AUCT satisfy the AUCT. As an application, we prove that certain simple C ∗ C^* -algebras which are locally type I are in fact isomorphic to simple AH-algebras. As another application, we show that a sequence of residually finite-dimensional C ∗ C^* -algebras which are asymptotically nuclear and which asymptotically satisfies the AUCT can be embedded into the same simple AF-algebra.