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Title:  On poperator spaces and their applications 
Author(s):  Lee, Jung Jin 
Director of Research:  Ruan, ZhongJin 
Doctoral Committee Chair(s):  Boca, Florin 
Doctoral Committee Member(s):  Ruan, ZhongJin; Junge, Marius; Erdogan, M. Burak 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  poperator space
pseudofunction algebra pseudomeasure algebra FigaTalamancaHerz algebra 
Abstract:  There have been a lot of research done on the relationship between locally compact groups and algebras associated with them. For example, Johnson proved that a locally compact group G is amenable if and only if the convolution algebra L1(G) is amenable as a Banach algebra, and Ruan showed that G is amenable if and only if the Fourier algebra A(G) of G is operator amenable. Motivated by Ruan's work, we want to study G through tools from poperator spaces. We _rst introduce the poperator space and various poperator space tensor products. We then study poperator space approximation property and poperator space completely bounded approximation property which are related to poperator space injective tensor product. We then apply these properties to the study of the pseudofunction algebra PFp(G), the pseudomeasure algebra PMp(G), and the Fig_aTalamancaHerz Algebra Ap(G). Especially we show that if G is discrete, the most of approximation properties for the reduced group C_algebra C_ _(G), the group von Neumann algebra V N(G), and the Fourier algebra A(G) (related to amenability, weak amenability, and approximation property of G) have natural panalogues for PFp(G), PMp(G), and Ap(G). With help of Herz's work, we also study the stability of these properties. Finally we discuss the properties Cp, C0p , and C00p which are natural panalogues of properties C, C0, and C00. 
Issue Date:  20100820 
URI:  http://hdl.handle.net/2142/16714 
Rights Information:  Copyright 2010 Jung Jin Lee 
Date Available in IDEALS:  20100820 
Date Deposited:  201008 
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Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois 
Dissertations and Theses  Mathematics