Haar measure and continuous representations of locally compact abelian groups
Volume 206 / 2011
Abstract
Let $\mathcal{L}(X)$ be the algebra of all bounded operators on a Banach space $X$, and let $\theta:G\rightarrow \mathcal{L}(X)$ be a strongly continuous representation of a locally compact and second countable abelian group $G$ on $X$. Set $\sigma^1(\theta(g)):=\{\lambda/|\lambda|\mid \lambda\in\sigma(\theta(g))\}$, where $\sigma(\theta(g))$ is the spectrum of $\theta(g)$, and let $\varSigma_\theta$ be the set of all $g\in G$ such that $\sigma^1(\theta(g))$ does not contain any regular polygon of $\mathbb{T}$ (by a regular polygon we mean the image under a rotation of a closed subgroup of the unit circle $\mathbb{T}$ different from $\{1\}$). We prove that $\theta$ is uniformly continuous if and only if $\varSigma_\theta$ is a non-null set for the Haar measure on $G$.
