Von Neumann techniques in C*-algebras

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.

The primary method of ensuring a pure mathematical research project of this nature has maximal impact in the short term is through wide scale dissemination in the academic community to ensure maximum reach of the results. Direct medium to longer term impact is obtained through skills and career development of the people involved in the project. In the longer term, the economic and social impact of research in pure mathematics is unquestionable.

In addition to publication of the project results in high profile journals and presentations in leading international conferences and workshops, two additional steps will be taken to ensure maximum academic impact, in particular with the next generation of researchers and researchers in neighbouring fields.

1. The investigators will continue to seek out opportunities to give expository masterclasses, aimed at PhD students and postdocs. They have an extensive track record in this direction, and the PI will present a mini course in May 2018.

2. A survey article will be written by the PI illustrating the connections between von Neumann algebras and C*-algebras, at a range of different levels. This will give researchers from outside the field a broad overview of the game changing developments of the last 5 years. This article will be targeted at the Bulletin of the American Mathematical Society, a high circulation journal regularly publishing excellent expository articles describing major advances to a general pure mathematical audience.

In addition to the impact on the wider research group in Glasgow, the project will have a major impact on the career of the postdoctoral researcher. The PI will hold monthly meetings with the postdoctoral researcher to provide support to their independent projects, and discuss career development (including supporting the researcher in developing funding and job applications). Further, the postdoctoral researcher will participate in the University of Glasgow's innovative early career development programme covering topics such as grant writing, presenting with impact, management and communication skills.

Additionally, the project will engage with two highly talented final year undergraduate students. Though working closely with the PI and postdoctoral researcher over an 8 week period, these undergraduate students will gain experience of how mathematical research is undertaken, develop teamwork, collaboration, writing and presentation skills as well as specialist knowledge. The impact of this intense period working with the PI and postdoctoral researcher will provide a significant benefit to these students next career steps, whether or not they pursue further mathematical research, as the skills that they will develop are widely applicable and very much in demand.

As a project in pure mathematics it is part of fundamental research which is pursued world wide. Whilst it is true that research of this nature has little immediate impact on everyday life, longer term it often has unexpected applications in science and economy. Examples include the use of abstract functional analysis in quantum physics, the use of number theory in data encryption modern science and economy, which in turn have enormous impact on society. This is unthinkable in it's current form without underpinning research in pure mathematics.

The EPSRC sponsored 2013 Deloitte report 'Measuring the Economic Benefits of Mathematical Science Research in the UK' estimated that there were `over 2.8 million individuals in direct employment due to mathematical science research in the UK' and `mathematical science research generated a direct gross value added in excess of £200 million' in 2010. So whilst it is difficult to predict the long term economic and societal impact of this concrete proposed research, it is vital that the UK continues to make long term strategic investments in pure mathematical research to maintain our expertise base and these strong economic benefits.