The Cuntz Semigroup and the Fine Structure of Nuclear C*-Algebras

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C*-algebras are norm-closed self-adjoint algebras of operators on Hilbert space. While these are fascinating and richly structured objects themselves, they also provide a natural framework to study connections between such widespread areas as functional analysis, algebra, topology, geometry, geometric group theory, and dynamical systems. Among all C*-algebras, nuclear ones are particularly well-behaved; they can be characterized in many ways, and are accessible to an abundance of techniques, often inspired by (algebraic) topology and geometry. A long-term project in the field is to classify nuclear C*-algebras by K-theoretic data. This is commonly referred to as Elliott programme; it is partially inspired by Connes' celebrated classification of injective factors in the 70s. The programme has seen tremendous progress in past decades, with a particular acceleration in the last 5 years. We now know that classifiability is related to dimension type properties, to tensorial absorption of strongly self-absorbing C*-algebras and to regularity properties of the classifying invariants. We also know that there are examples which cannot be distinguished by traditional K-theoretic invariants. Moreover, the current classification theory works best in the case of simple C*-algebras with an abundance of projections, and the technical difficulties in the non-simple case with few projections are substantial. There is growing body of evidence that a much finer invariant, the Cuntz semigroup, will be crucial to understand the fine structure of nuclear C*-algebras, and ultimately complete the classification problem. In this project we will systematically use Cuntz semigroup techniques to make progress on a range of ambitious problems in the classification programme. More specifically, the scientific aims are threefold. The first two parts below are of a fundamental and groundbreaking nature; the third part aims at applications and concrete classification results:(A) One of the main open problems in the area is to find range results for the Cuntz semigroup, i.e., determine which ordered abelian semigroups can occur as Cuntz semigroups of C*-algebras. The question seems to be extremely hard in general, but range results are indispensable for any successful classification theory, and the Elliott programme is no exception. (B) Many of the currently available classification results for nuclear C*-algebras follow a common pattern: an isomorphism of invariants is lifted to an invertible element of a bivariant theory using the Universal Coefficient Theorem (UCT); the result is then lifted to an isomorphism at the level of algebras. While by now it is clear that the Cuntz semigroup will play an important role as the classifying invariant in future classification results, there still is no bivariant version of it. We plan to develop such a bivariant Cuntz semigroup. We hope that this approach will also shed new light on the behaviour of the Cuntz semigroup with respect to small perturbations, and on the relations between the Cuntz semigroup and nuclearity. (C) In this part of the project we will focus on applications to concrete examples, and on the development of new classification theorems. In particular, we will compute the (bivariant) Cuntz semigroup for new classes of C*-algebras, e.g. for crossed products, for certain non-simple inductive limit C*-algebras, and for non-simple infinite C*-algebras; these results should also spur classification theorems for the same classes of C*-algebras. We will apply Cuntz semigroup techniques to study the fine structure of strongly self-absorbing C*-algebras. One of our motivations here is to make progress on the question whether the known strongly self-absorbing examples really are the only ones; this is related to one of the most important problems in the field, namely whether all nuclear C*-algebras satisfy the UCT.