  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.
