On iterated powers of positive definite functions

TL;DR

This paper proves that iterated powers of certain positive definite functions on locally compact groups tend to zero in weak* topology, and under irreducibility, they also converge to zero strongly on the group C*-algebra.

Contribution

It establishes convergence results for iterated powers of positive definite functions in both weak* and strong operator topologies, extending understanding of their asymptotic behavior.

Findings

Iterated powers of adapted positive definite functions converge to zero in weak* topology.

Irreducible positive definite functions' iterates converge to zero strongly on the group C*-algebra.

Results apply to functions with norm one in the Fourier--Stieltjes algebra.

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 $\|\rho\|_{B(G)}=1$, 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.