On iterated powers of positive definite functions

Abstract

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 $\Vert \rho {\Vert }_{B(G)}=1$, then the iterated powers $({\rho }^n)$ converge to zero in the weak* topology $\sigma (B(G),C^{\ast }(G))$. Moreover, if $\rho$ is irreducible, we prove that $({\rho }^n)$ as a sequence of u.c.p. maps on the group $C^{\ast }$-algebra converges to zero in the strong operator topology.