Topological and Algebraic Regularity Properties of Nuclear C*-Algebras

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A general principle of mathematical research is to study objects of high mathematical interest by 'classifying' them, that is, associate to each object another one, the 'invariant', which is of a simpler nature, yet retains a substantial amount of information about the initial object. One then seeks to compare the initial objects by studying their invariants. Such an approach is particularly satisfying if it is possible to view the objects and their invariants from a number of perspectives which are very different at first glance, yet are intimately related (sometimes for most subtle and surprising reasons).Our objects of interest are 'C*-algebras', which allow connections between such widespread areas as topology, group theory, analysis and dynamical systems. Since one cannot hope for a nontrivial classifying invariant for all C*-algebras, the fundamental questions then are: 1. What are reasonable classes of C*-algebras that are classifiable? 2. What are the suitable invariants?The classification of nuclear C*-algebras by K-theoretic invariants, commonly referred to as `Elliott program', has seen rapid development in recent years. On the one hand, ever larger subclasses of nuclear C*-algebras have been classified by their Elliott invariants; on the other hand, these results have made fruitful contact with other areas like the theory of dynamical systems or the Baum-Connes conjecture. Furthermore, the classification program has been a constant source of important new insights into the heart of the theory of C*-algebras itself. The scientific aim of the proposed research is threefold:A. For once, we are planning to extend the known classification results and to remove some technical and unnecessary constraints. A recent result of the PI suggests a concrete strategy of how to proceed in this direction. The problems along the way will be technically hard; we are hoping to bring together the expertise of leading researchers in the field, such as H.Lin, E.Kirchberg, N.C.Phillips, M.Dadarlat and M.Rordam. B. Furthermore, we plan to establish new applications of the existing (and upcoming) classification results to make them available to more examples from dynamical systems as well as graph theory. (For this task, it will be particularly important to remove certain technical constraints as indicated above.) We intend to hire a postdoctoral researcher with some expertise in a field allowing for applications of the classification program along these lines.C. Finally, we are aiming at a unified treatment of several scattered parts of the theory which are clearly related, but so far cannot be handled simultaneously in a conceptual way. We are confident that the new notions of noncommutative topological dimension and of D-stability can be connected in a satisfactory manner to make such a unification possible. The joint results of the PI with A.Toms and with M.Rordam have prepared ground for a conceptual treatment of purely finite C*-algebras, relating concepts called 'Z-stability', 'decomposition rank' and 'almost unperforated Cuntz semigroup'. We hope to substantially extend these results. Together with J.Zacharias I plan to establish a connection between purely finite and purely infinite C*-algebras via the so-called weak decompositon rank. This latter project is at its very beginnings, but has a high potential to allow for future applications. It also offers a number of possible starting points for PhD projects.At a strategic level, the project aims at extending the UK's internationally leading role in the theory of C*-algebras. While there are strong groups in other areas of the field (e.g. in Glasgow, Aberdeen, or Belfast), the development of the classification program in recent years has only marginally been driven by research groups from the UK. With the PI and J.Zacharias both based in Nottingham, we hope to establish an internationally leading centre in the classification program.