Noncommutative Lp-spaces associated with locally compact quantum groups

Locally compact quantum groups;Noncommutative Lp-spaces;Noncommutative harmonic analysis;Fourier transform;Fourier multipliers;Hausdorff-Young inequality

Cooney, Thomas J.

关键词: Locally compact quantum groups;    Noncommutative Lp-spaces;    Noncommutative harmonic analysis;    Fourier transform;    Fourier multipliers;    Hausdorff-Young inequality;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/16895/Cooney_Thomas.pdf?sequence=1&isAllowed=y

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来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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 ˆG. The Hausdorff–Young theorem states that if f ∈ L_p(G), where 1 ≤ p ≤ 2, then its Fourier transform F_p(f) belongs to L_q(ˆG) (where 1/p + 1/q = 1) and ||F_p(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 F_p : L_p (G) → L_q(^G) and showing that this Fourier transform satisfies the Hausdorff–Young inequality.Let G be a locally compact group. Then L_1(G) acts on L_p(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 L_p(VN(G)) has the operator space approximation property. Let G be a discrete Kac algebra with the approximation property. The aforementioned action of L_1(G) is used to show that L_p(ˆG) has the operator space approximation property. Similarly, if G is a weakly amenable discrete Kac algebra, then L_p(ˆG) has the completely bounded approximation property.