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

  • Group von Neumann algebras
  • Locally compact groups
  • Translation invariant means
  • Banach algebras
  • Translation operators

• Graduation date

  Fall 2010

• Type of Item

  Thesis

• Degree

  Doctor of Philosophy

• DOI

  https://doi.org/10.7939/R3HD7NZ4N

• License

  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.