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

TL;DR

This paper introduces an L^p-Fourier transform for locally compact quantum groups using non-commutative L^p-spaces, establishing its properties and relation to convolution products.

Contribution

It defines a new L^p-Fourier transform on quantum groups and characterizes its unique interpolation parameter, extending classical Fourier analysis to the quantum setting.

Findings

The L^p-Fourier transform is well-defined for 1 <= p <= 2.

The Fourier transform converts convolution into multiplication in the L^p-setting.

The interpolation parameter is uniquely determined by the Fourier transform.

Abstract

Using interpolation properties of non-commutative L^p-spaces associated with an arbitrary von Neumann algebra, we define a L^p-Fourier transform 1 <= p <= 2 on locally compact quantum groups. We show that the Fourier transform determines a distinguished choice for the interpolation parameter as introduced by Izumi. We define a convolution product in the L^p-setting and show that the Fourier transform turns the convolution product into a product.