  We prove that if $\rho$ is an adapted positive definite function in the
Fourier--Stieltjes algebra $B(G)$ of a locally compact group $G$ with
$\|\rho\|_{B(G)}=1$, then the iterated powers $(\rho^n)$ converge to zero in
the weak* topology $\sigma(B(G) , C^*(G))$. Moreover, if $\rho$ is irreducible,
we prove that $(\rho^n)$ as a sequence of u.c.p. maps on the group
$C^*$-algebra converges to zero in the strong operator topology.

Operator Algebras