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Journal of Operator Theory

Volume 64, Issue 1, Summer 2010  pp. 69-87.

Fourier transform on locally compact quantum groups

Authors Byung-Jay Kahng
Author institution: Department of Mathematics and Statistics, Canisius College, Buffalo, NY 14208, U.S.A.

Summary:  The notion of Fourier transform is among the most important tools in analysis, which has been generalized in abstract harmonic analysis to the level of abelian locally compact groups. The aim of this paper is to further generalize the Fourier transform. Motivated by some recent works by Van Daele in the multiplier Hopf algebra framework, and by using the Haar weights, we define here the (generalized) Fourier transform and the inverse Fourier transform, at the level of locally compact quantum groups. We then consider the analogues of the Fourier inversion theorem, Plancherel theorem, and the convolution product. Along the way, we also obtain an alternative description of the dual pairing map between a quantum group and its dual.

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