55 research outputs found
Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs
We study Gelfand pairs for locally compact quantum groups. We give an
operator algebraic interpretation and show that the quantum Plancherel
transformation restricts to a spherical Plancherel transformation. As an
example, we turn the quantum group analogue of the normaliser of SU(1,1) in
) together with its diagonal subgroup into a pair for which
every irreducible corepresentation admits at most two vectors that are
invariant with respect to the quantum subgroup. Using a -grading,
we obtain product formulae for little -Jacobi functions
The L^p-Fourier transform on locally compact quantum groups
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.Comment: 29 pages, to appear in the Journal of Operator Theor
Modular properties of matrix coefficients of corepresentations of a locally compact quantum group
We give a formula for the modular operator and modular conjugation in terms
of matrix coefficients of corepresentations of a quantum group in the sense of
Kustermans and Vaes. As a consequence, the modular autmorphism group of a
unimodular quantum group can be expressed in terms of matrix coefficients. As
an application, we determine the Duflo-Moore operators for the quantum group
analogue of the normaliser of SU(1,1) in ).Comment: 22 pages. To appear in Journal of Lie Theor
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