Banach algebra and operator space techniques in topological group theory

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There are two major strands lying behind our research. One is that of classical `harmonic analysis', in which functions are `separated into their periodic parts'. For example, a sound wave can be split into different frequencies, and once one is working in the `frequency domain' it is possible to process the signal-- for example, frequencies which the human ear cannot process can be removed, one of the ideas underpinning modern digital audio compression. This idea goes back nearly 300 hundred years to Fourier, who stated that `each function can be written as the infinite sum of trigonometric functions'. The developments of this idea lie at the base of much analysis in the subsequent centuries. In due course, in the 20th century, the subject of harmonic analysis became a huge and sophisticated subject; it considers functions and measures on general `locally compact groups' by inspecting the `Fourier transform' of these objects.The second strand behind our subject is that of bounded linear operators on Hilbert spaces; these operators generalize to infinitely many dimensions the idea of a matrix acting on a Euclidean space of finitely many dimensions. Hilbert spaces arise naturally in harmonic analysis: for example, a sound wave has infinitely many frequencies, each frequency giving rise to a dimension in `frequency space'. It is natural to expect that a sensible operation on frequency space will be linear-- the notion of adding sound waves, for example. The theory of operators on a Hilbert space also has numerous applications in the mathematical foundations of Quantum Mechanics, and historically arose in this area. These days this area of mathematics is refered to as the study of `operator algebras', and is a vast area of research with links across pure and applied mathematics.In the last 15--20 years a new link between operator algebras and harmonic analysis has arisen in the theory of `operator spaces'. This uses spaces of bounded linear operators on Hilbert spaces as a model for spaces as well as algebras. The natural maps between such objects are the `completely bounded operators'. Such maps seem to single out an important subclass of the collection of all bounded operators.Now this subject is being applied to modern harmonic analysis; it has already been shown that many results developed separately are really special cases of more general principles that have been discovered recently.It is the purpose of our workshop to bring together about 30 distinguished experts inboth aspects of our subject for a period of intensive study and lectures, intended for both the other experts and for graduate students and colleagues in other areas of mathematics. A number of UK-based people will attend these lectures.We expect existing collaborations between small groups of mathematicians to be consolidated and that new ones will form; these collaborations will generate, both within the workshop itself and in the future, some solutions to the open problems that we know of, and allow the participants to form new syntheses and approaches using ideas that they learn at the workshop. These new results will be disseminated in open access preprint servers and in international research journals.