The L^p-Fourier transform on locally compact quantum groups

November 15th, 2010, Analysis seminar (TU Delft).

Abstract

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. Using interpolation properties of non-commutative L^p-spaces associated with an arbitrary (not necessarily semi-finite) von Neumann algebra, we show how to define a L^p-Fourier transform on locally compact quantum groups. We show that convolution structures can be extended to the L^p setting and that the Fourier transform turns convolutions into products. We specialize the theory in case the von Neumann algebra of the quantum group is semi-finite or, moreover, type I.