Dual spaces and translation invariant means on group von Neumann algebras

Michael Yin-Hei Cheng ¹

¹ Department of Pure Mathematics University of Waterloo Waterloo, ON N2L 3G1, Canada

Studia Mathematica, Tome 223 (2014) no. 2, pp. 97-121

Résumé

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.

Détail

DOI : 10.4064/sm223-2-1

Keywords: locally compact group its dual space * set extreme points set normalized continuous positive definite functions early granirer rudin proved independently amenable discrete discrete only translation invariant means infty topologically invariant paper define study * translation operators via * investigate problem existence * translation invariant means which topologically invariant general properties * investigated