Haar measure and continuous representations of locally compact abelian groups

Jean-Christophe Tomasi¹

¹ IUFM Institut Universitaire de Formation des Maîtres Ancien Collège de Montesoro Avenue Paul Giacobbi 20600 Bastia, France

Studia Mathematica, Tome 206 (2011) no. 1, pp. 25-35

Résumé

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 $G$.

Détail

DOI : 10.4064/sm206-1-2

Keywords: mathcal algebra bounded operators banach space theta rightarrow mathcal strongly continuous representation locally compact second countable abelian group set sigma theta lambda lambda mid lambda sigma theta where sigma theta spectrum theta varsigma theta set sigma theta does contain regular polygon mathbb regular polygon mean image under rotation closed subgroup unit circle mathbb different prove theta uniformly continuous only varsigma theta non null set haar measure