Abstract. Say that a separable, unital C * -algebra D C is strongly selfabsorbing if there exists an isomorphism ϕ : D → D ⊗ D such that ϕ and id D ⊗ 1 D are approximately unitarily equivalent * -homomorphisms. We study this class of algebras, which includes the Cuntz algebras O 2 , O ∞ , the UHF algebras of infinite type, the Jiang-Su algebra Z and tensor products of O ∞ with UHF algebras of infinite type. Given a strongly self-absorbing C * -algebra D we characterise when a separable C * -algebra absorbs D tensorially (i.e., is D-stable), and prove closure properties for the class of separable D-stable C * -algebras. Finally, we compute the possible K-groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing C * -algebras.

We exhibit a counterexample to Elliott's classification conjecture for simple, separable, and nuclear C * -algebras whose construction is elementary, and demonstrate the necessity of extremely fine invariants in distinguishing both approximate unitary equivalence classes of automorphisms of such algebras and isomorphism classes of the algebras themselves. The consequences for the program to classify nuclear C * -algebras are far-reaching: one has, among other things, that existing results on the classification of simple, unital AH algebras via the Elliott invariant of K-theoretic data are the best possible, and that these cannot be improved by the addition of continuous homotopy invariant functors to the Elliott invariant.

Abstract. We prove that the Cuntz semigroup is recovered functorially from the Elliott invariant for a large class of C * -algebras. In particular, our results apply to the largest class of simple C * -algebras for which K-theoretic classification can be hoped for. This work has three significant consequences. First, it provides new conceptual insight into Elliott's classification program, proving that the usual form of the Elliott conjecture is equivalent, among Zstable algebras, to a conjecture which is in general substantially weaker and for which there are no known counterexamples. Second and third, it resolves, for the class of algebras above, two conjectures of Blackadar and Handelman concerning the basic structure of dimension functions on C * -algebras. We also prove in passing that the Cuntz-Pedersen semigroup is recovered functorially from the Elliott invariant for a large class of simple unital C * -algebras.

Abstract. We report on recent progress in the program to classify separable amenable C * -algebras. Our emphasis is on the newly apparent role of regularity properties such as finite decomposition rank, strict comparison of positive elements, and Z-stability, and on the importance of the Cuntz semigroup. We include a brief history of the program's successes since 1989, a more detailed look at the Villadsen-type algebras which have so dramatically changed the landscape, and a collection of announcements on the structure and properties of the Cuntz semigroup.

Contents 1. Introduction 1 2. The Cuntz semigroup: Definitions, historical origin and technical devices 2 3. Hilbert C*-modules. Kasparov's Theorem 17 4. The Category Cu 23 5. Classification of C * -algebras 41 References 54