Haar measure and continuous representations of locally compact abelian groups

Jean-Christophe Tomasi. "Haar measure and continuous representations of locally compact abelian groups." Studia Mathematica 206.1 (2011): 25-35. <http://eudml.org/doc/285777>.

@article{Jean2011,
abstract = {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 $Σ_\{θ\}$ 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 $Σ_\{θ\}$ is a non-null set for the Haar measure on G.},
author = {Jean-Christophe Tomasi},
journal = {Studia Mathematica},
keywords = {strongly and uniformly continuous representations of groups; Haar measure; null sets; essential spectrum},
language = {eng},
number = {1},
pages = {25-35},
title = {Haar measure and continuous representations of locally compact abelian groups},
url = {http://eudml.org/doc/285777},
volume = {206},
year = {2011},
}

TY - JOUR
AU - Jean-Christophe Tomasi
TI - Haar measure and continuous representations of locally compact abelian groups
JO - Studia Mathematica
PY - 2011
VL - 206
IS - 1
SP - 25
EP - 35
AB - 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 $Σ_{θ}$ 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 $Σ_{θ}$ is a non-null set for the Haar measure on G.
LA - eng
KW - strongly and uniformly continuous representations of groups; Haar measure; null sets; essential spectrum
UR - http://eudml.org/doc/285777
ER -