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Haar measure and continuous representations of locally compact abelian groups

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Tomasi J.-C.

Studia Mathematica

2011

tom 206

nr 1

25-35

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.

Ograniczanie wyników

1 Studia Mathematica

1 Tomasi J.-C.

1 2011

Wyniki wyszukiwania

Haar measure and continuous representations of locally compact abelian groups