Von Neumann techniques in C*-algebras

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The theory of operator algebras has its origins in quantum physics and the theory of unitary representations of locally compact groups. The area has many connections to other fields of mathematics with many more appearing in recent years. Deep structure results in the 1970s and new emerging applications to geometry and topology pioneered and condensed in the work of Kasparov have accelerated the development considerably. Today operator algebras has grown into a vast, attractive and very active area in modern mathematics.Traditionally, there are two main sub-areas of the field: von Neumann algebra theory and C*-algebra theory which are of fairly different flavour but in the end have striking similarities. Von Neumann algebras were first studied by Murray and von Neumann in the 1930's and 40's in connection with quantum physics. They are widely regarded as non-commutative measure spaces and a more akin to probability theory, more flexible than C*-algebras which were introduced by Gelfand and Naimark about a decade later. C*-algebras can be regarded as non-commutative topological spaces and their study is more akin to the study of spaces and geometric objects. For a long time, these sub-areas developed in parallel with limited direct connections between them. One of the major achievements in operator algebra theory is Connes' classification of amenable von Neumann algebras during the 1970's (completed by Haagerup in the 80's) which roughly means that these algebras can be reduced to a `list' of known examples. The Elliott programme launched in the late 80's has the ambitious goal to do something similar for C*-algebras: classify simple amenable C*-algebras by K-theory (and traces); here K-theory is a tool for classification of spaces from topology which applies to C*-algebras as well. This programme has seen dramatic recent progress and has now been solved for a definite class of algebras: those with finite nuclear dimension, a topological dimension concept analogous to the usual dimension of spaces. A major outstanding problem of the programme now is to find effective criteria to determine which C*-algebras have finite nuclear dimension, particularly in large classes of prominent examples for which classifiability is not yet known.A key theme emerging from recent major advances is the parallels between von Neumann algebra and C*-algebra theory. In particular many concepts used in the Connes-Haagerup classification of von Neumann algebras have analogues in the C*-world, not just at the conceptual level, but strong enough to be used in proofs. The major innovation of this proposal is to understand and develop these parallels fully and to apply this to the outstanding problem of identifying finite nuclear dimension.One of the most important classes we will consider are the crossed product algebras which are associated to dynamical systems (i.e. groups acting on spaces, such as irrational rotations of the circle). This is a major mathematical discipline in its own right, and the strong connections to operator algebras date back to the work of Murray and von Neumann. Measurable dynamics correspond to von Neumann crossed products whereas continuous dynamics to C*-crossed products. The latter provide indispensable guiding examples of simple amenable C*-algebras which have and are being studied intensively. Tremendous progress has been made recently for actions of certain groups like the integers, which are relatively small (in a coarse sense). We aim to develop new methods, which work much more generally, and allow us to completely characterise when simple crossed product C*-algebras have finite nuclear dimension. To allow this to be widely used, the characterisation we seek will be entirely dynamical in nature, and readily checkable in concrete examples.