|
Gelfand pairs of quantum groups
August 9th, 2011, Analysis seminar (Monash University, Melbourne).
Abstract
Compact quantum groups were introduced in an operator algebraic setting by Woronowicz in 1987. They appear as certain deformations of the algebra of functions on a (compact) Lie group. Using the techniques from operator algebras, many results from harmonic analysis could be interpreted in this quantum setting. Around 2000 a satisfactory, von Neumann algebraic definition for locally compact quantum groups was introduced by Kustermans and Vaes, allowing to develop the theory even further for non-compact examples.
The goal of this talk is twofold. In the first part, I give an introduction to operator algebraic quantum groups. We give the classical case of functions on a locally compact group and a deformed example, namely SU_q(2). Meanwhile, we focus on the machinery needed to define Fourier transforms. In the second part I sketch how a spherical Fourier transformation can be defined on pairs of locally compact quantum group that are 'almost Gelfand pairs'.
L^p-Fourier transforms on (quantum) groups
August 10th, 2011, Analysis seminar (Monash University, Melbourne).
Abstract
Abstract: We start with an introduction to non-commutative L^p-spaces associated with an arbitrary (non-semi-finite) von Neumann algebra. Next, we will tell how Fourier transforms of quantum groups (so in particular on any locally compact group) can be defined. Classical results translate: for example, we have an inversion theorem and convolutions are turned into products by the Fourier transform.
|
|