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

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.

Pure mathematics have proven to be indispensable for essentially all disciplines in science and engineering, and for the development of an abundance of technologies. For example, methods from number theory are nowadays used for data encryption, and digitalisation of signals relies heavily on ideas from functional analysis, in particular the theory of Hilbert spaces. It is remarkable that the groundbreaking mathematics for these applications resulted from merely curiosity driven fundamental research, with the actual applications (as well as enormous economic and societal impact) some fifty or hundred years ahead. While such a long-term impact is impossible to predict, let alone to plan, our project is rooted in a field which has many connections to other fields and even disciplines, such as dynamical systems, geometry, and mathematical physics. The requested travel funds, and the Masterclass and Workshop will obviously generate direct economic impact. Our project will increase the UK's standing in contemporary fundamental research, and hence its attractivity as a place to study and do research; this will generate an indirect societal impact. Our results will be published as a series of research articles. These will be submitted to peer-reviewed high-quality international journals; they will also be posted on the arXiv preprint server for immediate and public availability. We will make an effort to publish survey articles/research announcements in high-impact multidisciplinary journals (this approach has already been successful; see the PNAS articles by the PI et. al.). The results will furthermore be disseminated by the involved researchers at international conferences and workshops (including the proposed Masterclass and Workshop). These will also serve as networking events and establish new collaborations. The project will have its own webpage to communicate new developments and activities, this will in particular promote and support the proposed Masterclass and Workshop. Together with the PI, the CoI, the PostDoc, the PhD student and visiting and visited researchers, there will at least 16 national and international researchers be involved in the project. Besides the obvious impact on the careers of these, we also expect a strong multiplying effect, since we expect our results to be a basis for future research projects, which will in turn be promoted by the involved researchers. In particular, there will be a number of sidetracks which are highly suitable as starting points for PhD theses. We expect the Masterclass and Workshop to have similar multiplying effects. (For example, the PI's First Grant has had a positive impact on the PhD project of Bhishan Jacelon from Glasgow. Jacelon is also organising an event in Glasgow aimed at UK and international PhD students, which is linked to and which will directly precede the LMS Regional Meeting and Workshop in Nottingham in September 2010.) The expected series of high-quality publications, and the new research and collaboration opportunities will have a substantial positive impact especially on the CVs and careers of the involved junior researchers. We will ensure that the PostDoc and PhD student get the opportunity to attend international conferences, hence promote their results (and themselves) to a wide audience. The proposed workshop will bring together all researchers involved in the project with other experts in the field; the masterclass is of particular importance, since it will be aimed at interested PhD students from a broad range of fields; it will hence promote our project to future experts in- and outside our own area.