Haar measure and continuous representations of locally compact abelian groups

• Autorzy: Jean-Christophe Tomasi
• Języki publikacji: EN

Abstrakty

EN

Let 𝓛(X) be the algebra of all bounded operators on a Banach space X, and let θ: G → 𝓛(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 𝕋 (by a regular polygon we mean the image under a rotation of a closed subgroup of the unit circle 𝕋 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.

Kategorie tematyczne

• 22A25: Representations of general topological groups and semigroups
• 47D03: Groups and semigroups of linear operators
• 47A10: Spectrum, resolvent

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2011
• Tom: 206
• Numer: 1
• Strony: 25-35

Daty

wydano

2011

Twórcy

autor

Jean-Christophe Tomasi

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

Identyfikatory

DOI

10.4064/sm206-1-2