The Lp-Fourier transform on locally compact quantum groups
Locally compact (l.c.) quantum groups have been introduced in a von Neumann algebraic setting by Stefaan Vaes and Johan Kustermans in 2000. One of their main motivations was to generalize the Pontrjagin duality theorem for abelian l.c. groups. So to every l.c. quantum group one can associate a dual l.c. quantum group, such that the double dual is the l.c. quantum group itself. It is known that many other aspects of harmonic analysis have a suitable interpretation in the quantum group setting.
In this talk I will give a short introduction to quantum groups, focussing on the von Neumann algebraic definition. We will sketch how a L^p-Fourier transform can be realized on these quantum groups, what problems arise and how they shed light on interpolation properties of von Neumann algebras.