# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Haar measure and continuous representations of locally compact abelian groups

### Tom 206 / 2011

Studia Mathematica 206 (2011), 25-35 MSC: 47A10, 47D03, 22A25. DOI: 10.4064/sm206-1-2

#### Streszczenie

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 $G$.

#### Autorzy

• Jean-Christophe TomasiIUFM Institut Universitaire de Formation des Maîtres
Ancien Collège de Montesoro
Avenue Paul Giacobbi
20600 Bastia, France
e-mail

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