Uniformly ergodic probability measures

Abstract

Let $G$ be a locally compact group and $\mu$ be a probability measure on $G$. We consider the convolution operator $\lambda_1(\mu)\colon L_1(G)\to L_1(G)$ given by $\lambda_1(\mu)f=\mu \ast f$ and its restriction $\lambda_1^0(\mu)$ to the augmentation ideal $L_1^0(G)$. Say that $\mu$ is uniformly ergodic if the Ces\`aro means of the operator $\lambda_1^0(\mu)$ converge uniformly to 0, that is, if $\lambda_1^0(\mu)$ is a uniformly mean ergodic operator with limit 0 and that $\mu$ is uniformly completely mixing if the powers of the operator $\lambda_1^0(\mu)$ converge uniformly to 0. We completely characterize the uniform mean ergodicity of the operator $\lambda_1(\mu)$ and the uniform convergence of its powers and see that there is no difference between $\lambda_1(\mu)$ and $\lambda_1^0(\mu) $ in this regard. We prove in particular that $\mu$ is uniformly ergodic if and only if $G$ is compact, $\mu$ is adapted (its support is not contained in a proper closed subgroup of $G$) and 1 is an isolated point of the spectrum of $\mu$. The last of these three conditions is actually equivalent to $\mu$ being spread-out (some convolution power of $\mu$ is not singular). The measure $\mu$ is uniformly completely mixing if and only if $G$ is compact, $\mu$ is spread-out and the only unimodular value of the spectrum of $\mu$ is 1.