IDEALS Home University of Illinois at Urbana-Champaign logo The Alma Mater The Main Quad

Noncommutative Lp-spaces associated with locally compact quantum groups

Show full item record

Bookmark or cite this item: http://hdl.handle.net/2142/16895

Files in this item

File Description Format
PDF Cooney_Thomas.pdf (388KB) (no description provided) PDF
Title: Noncommutative Lp-spaces associated with locally compact quantum groups
Author(s): Cooney, Thomas J.
Director of Research: Ruan, Zhong-Jin
Doctoral Committee Chair(s): Boca, Florin
Doctoral Committee Member(s): Ruan, Zhong-Jin; Junge, Marius; Fima, Pierre
Department / Program: Mathematics
Discipline: Mathematics
Degree Granting Institution: University of Illinois at Urbana-Champaign
Degree: Ph.D.
Genre: Dissertation
Subject(s): Locally compact quantum groups Noncommutative Lp-spaces Noncommutative harmonic analysis Fourier transform Fourier multipliers Hausdorff-Young inequality
Abstract: 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.
Issue Date: 2010-08-20
URI: http://hdl.handle.net/2142/16895
Rights Information: Copyright 2010 Thomas J. Cooney
Date Available in IDEALS: 2010-08-20
Date Deposited: 2010-08
 

This item appears in the following Collection(s)

Show full item record

Item Statistics

  • Total Downloads: 234
  • Downloads this Month: 5
  • Downloads Today: 0

Browse

My Account

Information

Access Key