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