THE ISOLATED POINTS OF $\hat{G}_\rho$ AND THE $w^\ast$-STRONGLY EXPOSED POINTS OF $P_{\rho}(G)_0$

Abstract

Let $G$ be a separable locally compact group and let $\hat{G}$ be its dual space with Fell's topology. It is well known that the set $P(G)$ of continuous positive-definite functions on $G$ can be identified with the set of positive linear functionals on the group $C^*$ -algebra $C^*(G)$ . We show that if $\pi$ is discrete in $\hat{G}$ , then there exists a nonzero positive-definite function $\phi_{\pi}$ associated with $\pi$ such that $\phi_{\pi}$ is a $w^*$ -strongly exposed point of $P(G)_0$ , where $P(G)_0=\{f\in p(G):f(e)\le 1\}$ . Conversely, if some nonzero positive-definite function $\phi_\pi$ associated with $\pi$ is a $w^*$ -strongly exposed point of $P(G)_0$ , then $\pi$ is isolated in $\hat{G}$ . Consequently, $G$ is compact if and only if, for every $\pi\in\hat{G}$ , there exists a nonzero positive-definite function associated with $\pi$ that is a $w^*$ -strongly exposed point of $P(G)_0$ . If, in addition, $G$ is unimodular and $\pi\in\hat{G}_\rho$ , then $\pi$ is isolated in $\hat{G}_{\rho}$ if and only if some nonzero positive-definite function associated with $\pi$ is a $w^*$ -strongly exposed point of $P_{\rho}(G)_0$ , where $\rho$ is the left regular representation of $G$ and $\hat{G}_{\rho}$ is the reduced dual space of $G$ . We prove that if $B_{\rho}(G)$ has the Radon–Nikodym property, then the set of isolated points of $\hat{G}_{\rho}$ (so square-integrable if $G$ is unimodular) is dense in $\hat{G}_{\rho}$ . It is also proved that if $G$ is a separable SIN-group, then $G$ is amenable if and only if there exists a closed point in $\hat{G}_{\rho}$ . In particular, for a countable discrete non-amenable group $G$ (for example the free group $F_2$ on two generators), there is no closed point in its reduced dual space $\hat{G}_{\rho}$ .