Extreme points of the unit ball of the Fourier-Stieltjes algebra

Let $G$ be a locally compact group. Among other things, we proved in this paper that for an IN-group $G$, the extreme points of the unit ball of the Fourier-Stieltjes algebra $B(G)$ are not in the Fourier algebra $A(G)$ if and only if $G$ is non-compact, or equivalently, there is no irreducible representation of $G$ which is quasi-equivalent to a subrepresentation of the left regular representation of $G$ if and only if $G$ is non-compact. This result is a non-commutative version of the following well known result: For any locally compact group $\widehat G$, the extreme points of the unit ball of the measure algebra $M(\widehat G)$ are not in the group algebra $L^{1}(\widehat G)$ if and only if $\widehat G$ is non-discrete. On the other hand, we also showed that if $B(G)$ has the RNP, then there are extreme points of the unit ball of $B(G)$ that are in $A(G)$. Since it is well known there are non-compact locally compact group $G$ for which $B(G)$ has the RNP, there exist non-compact locally compact groups $G$ where extreme points of the unit ball of $B(G)$ can be in $A(G)$. This shows that the condition $G$ be an IN-group cannot be entirely removed.