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Journal of Operator Theory

Volume 69, Issue 1, Winter 2013  pp. 161-193.

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

Authors Martijn Caspers
Author institution: IMAPP, Radboud Universiteit, Nijmegen, 6525 AJ, The Netherlands

Summary:  Using interpolation properties of non-commutative $L^p$-spaces associated with an arbitrary von Neumann algebra, we define an $L^p$-Fourier transform $1 \leqslant p \leqslant 2$ 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.

Keywords:  locally compact quantum groups, Fourier transform, interpolation spaces

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