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Dual spaces and translation invariant means on group von Neumann algebras

Volume 223 / 2014

Michael Yin-Hei Cheng Studia Mathematica 223 (2014), 97-121 MSC: Primary 43A07, 43A22, 46J10; Secondary 43A40, 46J30. DOI: 10.4064/sm223-2-1


Let $G$ be a locally compact group. Its dual space, $G^*$, is the set of all extreme points of the set of normalized continuous positive definite functions of $G$. In the early 1970s, Granirer and Rudin proved independently that if $G$ is amenable as discrete, then $G$ is discrete if and only if all the translation invariant means on $L^\infty (G)$ are topologically invariant. In this paper, we define and study $G^*$-translation operators on ${\rm VN}(G)$ via $G^*$ and investigate the problem of the existence of $G^*$-translation invariant means on ${\rm VN}(G)$ which are not topologically invariant. The general properties of $G^*$ are also investigated.


  • Michael Yin-Hei ChengDepartment of Pure Mathematics
    University of Waterloo
    Waterloo, ON N2L 3G1, Canada

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