Let (X) be the algebra of all bounded operators on a Banach space X, and let θ: G → (X) be a strongly continuous representation of a locally compact and second countable abelian group G on X. Set σ¹(θ(g)): = λ/|λ| | λ ∈ σ(θ(g)), where σ(θ(g)) is the spectrum of θ(g), and let be the set of all g ∈ G such that σ¹(θ(g)) does not contain any regular polygon of (by a regular polygon we mean the image under a rotation of a closed subgroup of the unit circle different from 1). We prove that θ is uniformly continuous if and only if is a non-null set for the Haar measure on G.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm206-1-2,
author = {Jean-Christophe Tomasi},
title = {Haar measure and continuous representations of locally compact abelian groups},
journal = {Studia Mathematica},
volume = {204},
year = {2011},
pages = {25-35},
zbl = {1258.47007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm206-1-2}
}
Jean-Christophe Tomasi. Haar measure and continuous representations of locally compact abelian groups. Studia Mathematica, Tome 204 (2011) pp. 25-35. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm206-1-2/