Matt Daws (Leeds) - "Analysis in 'non-commutative' mathematics"

Abstract

I will give a survey style talk. Geometry (or more to my tastes, topology) studies sets with additional structures, such as topologies, metrics, manifold structures etc. Often looking at algebras of (complex-valued) functions on such spaces captures the underlying space - of interest here is the Gelfand Theorem which tells us that commutative C*-algebras are precisely the algebras of continuous functions (vanishing at infinity) on locally compact spaces. So commutative C*-algebras and locally compact spaces "are the same" (I shall try to be a little precise here). Thus we can think of non-commutative C*-algebras as being "non-commutative spaces"; one has to have a little "taste" of course - not every naive generalisation will be profitable. I will discuss Woronowicz's notion of a "compact quantum group". These are a, on the face of it rather naive, generalisation of compact groups; but we shall see that a simple set of axioms leads to a structure every bit as rich as the classical notion of a compact group. I will give some examples, making links with Fourier analysis, for example.