Ergodic properties of convolution operators in group algebras
Volume 165 / 2021
Colloquium Mathematicum 165 (2021), 321-340
MSC: Primary 46Hxx, 43A10, 43A20, 43A25, 43A45; Secondary 22D15, 37A30
DOI: 10.4064/cm8214-6-2020
Published online: 1 February 2021
Abstract
Let $G$ be a locally compact abelian group and let $L^{1} ( G ) $ and $M ( G ) $ be respectively the group algebra and the convolution measure algebra of $G$. For $\mu \in M ( G )$, let $T_{\mu }f=\mu \ast f$ be the convolution operator on $L^{1} ( G ) $. A measure $\mu \in M ( G ) $ is said to be power bounded if $\sup _{n\geq 0} \Vert \mu ^{n} \Vert _{1} \lt \infty $, where $\mu ^{n}$ denotes the $n$th convolution power of $\mu $. We study some ergodic properties of the convolution operator $T_{\mu }$ in the case when $\mu $ is power bounded. We also present some results concerning almost everywhere convergence of the sequence $ \{ T_{\mu }^{n}f \} $ in $L^{1} ( G )$.
