Noncommutative Lp-spaces associated with locally compact quantum groups

Results from abstract harmonic analysis are extended to locally compact quantum groups by considering the noncommutative Lp-spaces associated with the locally compact quantum groups.;Let G be a locally compact abelian group with dual group G. The Hausdorff-Young theorem states that if f ∈ Lp(G), where 1 ≤ p ≤ 2, then its Fourier transform Fp (f) belongs to Lq( G) (where 1p+1q = 1) and ∥ Fp (f)∥q ≤ ∥f∥ p. Kunze and Terp extended this to unimodular and locally compact groups, respectively. We further generalize this result to an arbitrary locally compact quantum group G by defining a Fourier transform Fp : Lp( G ) → Lq( G&d4; ) and showing that this Fourier transform satisfies the Hausdorff-Young inequality.;Let G be a locally compact group. Then L 1(G) acts on Lp( G) by convolution. We extend this result to Kac algebras and also discuss an operator space version of this result. Ruan and Junge showed that if G is a discrete group with the approximation property, then Lp(V N(G)) has the operator space approximation property. Let G be a discrete Kac algebra with the approximation property. The aforementioned action of L1( G ) is used to show that Lp( G&d4; ) has the operator space approximation property. Similarly, if G is a weakly amenable discrete Kac algebra, then Lp( G&d4; ) has the completely bounded approximation property.