Spectral theory of group representations and their nonstandard hull
type: Item
instance of: scholarly article
title: Spectral theory of group representations and their nonstandard hull
zbMATH Open document ID: 0568.22005
DOI: 10.1007/BF02761165
author: Manfred P. H. Wolff
published in: Israel Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: Let G denote a locally compact abelian group and let U be a bounded strongly continuous representation of G on the Banach space E. We introduce the notion of the Riesz part \(R\sigma\) (U) of the Arveson spectrum \(\sigma\) (U) of U. The representation U is called R-compact if every bounded subset \(C\subset E\) satisfying \(\lim_{t\to e}\sup \{\| U_ tx-x\|:x\in C\}=0\) is relatively compact. Then the following assertions are equivalent (i) \(\sigma (U)=R\sigma (U);\) (ii) U is R- compact; (iii) For all \(f\in L^ 1(G)\) the associated operator \(U_ f\) is compact. This theorem is applied to the complete characterization of R-compact representations on Banach lattices, which generalizes results of \textit{G. Greiner} [Über das Spektrum stark stetiger Halbgruppen positiver Operatoren, Diss. Tübingen (1980; Zbl 0475.47025)] as well as of \textit{H. Uhlig} [Derivationen und Verbandshalbgruppen, Diss. Tübingen (1979; Zbl 0453.47019)] even in the case of \(G={\mathbb{R}}\). As an important tool for the study of the spectrum we introduce the nonstandard hull of an arbitrary group representation which facilitates all proofs of known results as well as the new ones represented here.
Mathematics Subject Classification ID: 22D12
Mathematics Subject Classification ID: 47D03
zbMATH DE Number: 3906765
zbMATH Keywords: locally compact abelian group
zbMATH Keywords: bounded strongly continuous representation
zbMATH Keywords: Riesz part
zbMATH Keywords: Arveson spectrum
zbMATH Keywords: R-compact representations
zbMATH Keywords: nonstandard hull
MaRDI profile type: Publication
cites work: Q3930261
cites work: On groups of automorphisms of operator algebras
cites work: Q4743311
cites work: Upper and Lower Fredholm Spectra
cites work: A Quotient of Ultrapowers of Banach Spaces and Semi-Fredholm Operators
cites work: Q4180808
cites work: Über das Spektrum positiver Generatoren
cites work: Harmonic analysis based on certain commutative Banach algebras
cites work: Three spectral notions for representations of commutative Banach algebras
cites work: On the spectrum of a one-parameter strongly continous representation.
cites work: Applications of almost periodic compactifications
cites work: Q3931940
cites work: Zur Perron-Frobenius-Theorie stark stetiger Halbgruppen
cites work: A Perron Frobenius theory for representations of locally compact Abelian groups
cites work: Q3923136
cites work: Q3205756
cites work: Q5608624
cites work: Q5644185
cites work: On representations with a separable spectrum
cites work: Hyperfinite Extensions of Bounded Operators on a Separable Hilbert Space
cites work: Abstract dynamical systems with an application to operators with discrete spectrum
cites work: Q5796220
cites work: On Norm-Continuity and Compactness of Spectrum.
cites work: Q3205815
cites work: Q4049667
cites work: Q4104164
cites work: Q3900547
cites work: Q3952692
cites work: Q4774550
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description: scientific article; zbMATH DE number 3906765