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Abstract

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.