On The Classification of Nuclear C * ‐Algebras

Abstract

Two of the most influential works on C-algebras from the mid-seventies ‐ Brown, Douglas and Fillmore’s [6] and Elliott’s [21] ‐ both contain uniqueness and existence results in the now standard sense which we shall outline below. These papers served as keystones for two separate theories ‐ KK-theory and the classification program ‐ which for many years parted ways with only moderate interaction. But with this common origin in mind, it is not surprising that recent years have seen a fruitful interaction which has been one of the main engines behind rapid progress in the classification program. In the present paper we take this interaction even further. Combining a concept of quasidiagonality for representations and a new characterization of equivalence in the Cuntz picture of KK-theory, we achieve a general uniqueness result. And from the quasidiagonality notion we derive a corresponding general existence result by comparing well-known realizations of the KK-groups. These results are then employed to obtain new classification results for certain classes of quasidiagonal C-algebras introduced by H. Lin. An important novel feature of these classes is that they are defined by a certain local approximation property, rather than by an inductive limit construction. We emphasize that our fundamental uniqueness result does not depend on the universal coefficient theorem (UCT), nor on nuclearity of the C-algebras involved. On the other hand, when we refine the uniqueness result for use in classification, we need to gradually add such conditions on the C-algebras. It is to be expected that requirements of this type are necessary, but we have found that holding them back as long as possible leads to more conceptual proofs while clarifying the role of the UCT and nuclearity in classification. Further, since uniqueness results answer fundamental questions about KK- and K-theory, they are of interest in themselves. We have pursued this theme in [14], and defer to this paper the proof of the new characterization of KK-theory which lies behind our uniqueness results. Our existence and uniqueness results are in the spirit of the classical Ext-theory from [6]. The main complication overcome in the paper is to control the stabilization which is necessary when one works with finite C-algebras. In the infinite case, where programs of this type have already been successfully carried out, stabilization is unnecessary. Yet, our methods are sufficiently versatile to allow us to reprove, from a handful of basic results, the classification of purely infinite nuclear C-algebras of Kirchberg and Phillips.