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# Translation operators on group von Neumann algebras and Banach algebras related to locally compact groups

• Author / Creator
Cheng, Yin-Hei
• Let $G$ be a locally compact group, $G^*$ be the set of all extreme points of the set of normalized continuous positive definite functions of $G$ and $a(G)$ be the closed subalgebra generated by $G^*$ in $B(G)$. When $G$ is abelian, $G^*$ is the set of dirac measures of the dual group of $G$. The general properties of $G^*$ are investigated in this thesis. We study the properties of $a(G)$, particularly on its spectrum. We also define translation operators on $VN(G)$ via $G^*$ and investigate the problem of the existence of translation means on $VN(G)$ which are not topological invariant. Lastly, we define reflexivity of subgroups of $G$ by using $G^*$, and show that a subgroup $H$ is reflexive if and only if $G$ had $H$-separation property. If $G$ is abelian, there is correspondence between closed subgroups of $G$ and closed subgroups of the dual group $\hat{G}$. We generalize this result to the class of groups having separation property.

• Subjects / Keywords
2010-11
• Type of Item
Thesis
• Degree
Doctor of Philosophy
• DOI
https://doi.org/10.7939/R3HD7NZ4N
This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
• Language
English
• Institution
University of Alberta
• Degree level
Doctoral
• Department
• Department of Mathematical and Statistical Sciences
• Supervisor / co-supervisor and their department(s)
• Lau, Anthony To-Ming (Mathematics and Statistical Sciences)
• Examining committee members and their departments
• Al-Hussein, Mohamed (Construction Engineering and Management)
• Troitsky, Vladimir G. (Mathematics and Statistical Sciences)
• Runde, Volker (Mathematics and Statistical Sciences)
• Dai, Feng (Mathematics and Statistical Sciences)
• Derighetti, Antoine (Institut de Mathématiques, Université de Lausanne)