Positive definite functions as uniformly ergodic multipliers of the Fourier algebra

Abstract

Let G be a locally compact group and let $\varphi$ be a positive definite function on G with $\varphi (e)=1$. This function defines a multiplication operator $M_{\varphi }$ on the Fourier algebra $A(G)$ of $G$. The aim of this paper is to classify the ergodic properties of the operators $M_{\varphi }$, focusing on several key factors, including the subgroup $H_{\varphi }=\{x\in G:\varphi (x)=1\}$, the spectrum of $M_{\varphi }$, or how ``spread-out'' a power of $M_{\varphi }$ can be. We show that the multiplication operator $M_{\varphi }$ is uniformly mean ergodic if and only if $H_{\varphi }$ is open and 1 is not an accumulation point of the spectrum of $M_{\varphi }$. Equivalently, this happens when some power of $\varphi$ is not far, in the multiplier norm, from a function supported on finitely many cosets of $H_{\varphi }$. Additionally, we show that the powers of $M_{\varphi }$ converge in norm if, and only if, the operator is uniformly mean ergodic and $H_{\varphi }=\{x\in G:|\varphi (x)|=1\}$.

21 pages