No abstract

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.

Various subsets of the tracial state space of a unital C * -algebra are studied. The largest of these subsets has a natural interpretation as the space of invariant means. II 1 -factor representations of a class of C * -algebras considered by Sorin Popa are also studied. These algebras are shown to have an unexpected variety of II 1 -factor representations. In addition to developing some general theory we also show that these ideas are related to numerous other problems in operator algebras.To my big, beautiful family -on both sides of the Pacific. ContentsChapter 1. Introduction Chapter 2. Notation, definitions and useful facts Chapter 3. Amenable traces and stronger approximation properties 3.1. Characterizations of amenable traces 3.2. Uniform amenable traces 3.3. Quasidiagonal traces 3.4. Locally finite dimensional traces 3.5. Miscellaneous remarks and permanence properties Chapter 4. Examples and special cases 4.1. Discrete groups 4.2. Nuclear and WEP C * -algebras 4.3. Locally reflexive, exact and quasidiagonal C * -algebras 4.4. Type I C * -algebras 4.5. Tracially AF C * -algebras Chapter 5. Finite representations 5.1. II 1 -factor representations of some universal C * -algebras 5.2. Elliott's intertwining argument for II 1 -factors 5.3. II 1 -factor representations of Popa Algebras Chapter 6. Applications and connections with other areas 6.1. Elliott's classification program 6.2. Counterexamples to questions of Lin and Popa 6.3. Connes' embedding problem 6.4. Amenable traces and numerical analysis 6.5. Amenable traces and obstructions in K-homology 6.6. Stable finiteness versus quasidiagonality 6.7. Questions Bibliography vii 1 At this point one may wonder if this notion depends on the particular choice of representation A ⊂ B(H). It doesn't, as we will observe later.

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.

Let Γ be a discrete group. To every ideal in ∞ (Γ), we associate a C * -algebra completion of the group ring that encapsulates the unitary representations with matrix coefficients belonging to the ideal. The general framework we develop unifies some classical results and leads to new insights. For example, we give the first C * -algebraic characterization of a-T-menability; a new characterization of property (T); new examples of 'exotic' quantum groups; and, after extending our construction to transformation groupoids, we improve and simplify a recent result of Douglas and Nowak ['Hilbert C * -modules and amenable actions ', Studia Math. 199 (2010) 185-197].here, s * μ is the translate of the measure μ by the group element s, and ds * μ/dμ is the Radon-Nikodym derivative. Douglas and Nowak show that ifρ is integrable, or if ρ is non-zero, then the group Γ is a-T-menable. They ask whether amenability of Γ follows from either of these conditions. In this note, we shall prove that this is indeed the case. See Corollary 5.11 and the surrounding discussion.Our initial result led us to the following question: if one wishes to conclude a-T-menability of Γ, then what are the appropriate hypotheses? To answer this question, we introduce appropriate completions of the group ring of G, and of the convolution algebra C c (X G) in the case of