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.

Abstract. We prove that faithful traces on separable and nuclear C * -algebras in the UCT class are quasidiagonal. This has a number of consequences. Firstly, by results of many hands, the classification of unital, separable, simple and nuclear C * -algebras of finite nuclear dimension which satisfy the UCT is now complete. Secondly, our result links the finite to the general version of the Toms-Winter conjecture in the expected way and hence clarifies the relation between decomposition rank and nuclear dimension. Finally, we confirm the Rosenberg conjecture: discrete, amenable groups have quasidiagonal C * -algebras.

We introduce the concept of finitely coloured equivalence for unital * -homomorphisms between C * -algebras, for which unitary equivalence is the 1-coloured case. We use this notion to classify *homomorphisms from separable, unital, nuclear C * -algebras into ultrapowers of simple, unital, nuclear, Z-stable C * -algebras with compact extremal trace space up to 2-coloured equivalence by their behaviour on traces; this is based on a 1-coloured classification theorem for certain order zero maps, also in terms of tracial data.As an application we calculate the nuclear dimension of non-AF, simple, separable, unital, nuclear, Z-stable C * -algebras with compact extremal trace space: it is 1. In the case that the extremal trace space also has finite topological covering dimension, this confirms the remaining open implication of the Toms-Winter conjecture. Inspired by homotopy-rigidity theorems in geometry and topology, we derive a "homotopy equivalence implies isomorphism" result for large classes of C * -algebras with finite nuclear dimension.

Abstract. We introduce the decomposition rank, a notion of covering dimension for nuclear C * -algebras. The decomposition rank generalizes ordinary covering dimension and has nice permanence properties; in particular, it behaves well with respect to direct sums, quotients, inductive limits, unitization and quasidiagonal extensions. Moreover, it passes to hereditary subalgebras and is invariant under stabilization. It turns out that the decomposition rank can be finite only for strongly quasidiagonal C * -algebras and that it is closely related to the classification program.

No abstract