Unitary representations and compact groups
type: Item
instance of: scholarly article
title: Unitary representations and compact groups
zbMATH Open document ID: 0738.43009
DOI: 10.1007/BF01190120
published in: Archiv der Mathematik
publication date: 1992-01-01T00:00:00Z
review text: We study relations between compact tall groups \(G\) (i.e. the number of irreducible representations of \(G\) of dimension \(n\) is finite for all \(n\in\mathbb{N}\)) and groups such that the left regular representations restricted to the orthocomplement of the constant functions weakly contain the trivial representation (we write \(1\prec\pi_ G\)). For Lie groups \(G\) is not tall if and only if \(1\prec \pi_ G\) [see \textit{C. Chou}, Trans. Am. Math. Soc. 317, 229-253 (1990; Zbl 0694.43007)]. This follows immediately from results of \textit{G. A. Margulis} [Monatsh. Math. 90, 233-236 (1980; Zbl 0425.43001)] and \textit{V. G. Drinfel'd} [Funkts. Anal. Prilozh. 18, No. 3, 77 (1984; Zbl 0576.28019)] who considered the case of simple groups. If \(G\) is not tall, then it is shown that \(1\prec\pi_ G\). The converse is not true, as stated by \textit{C. Chou} [loc. cit., Theorem 3.5] but the proof given there is not correct and the statements of Lemma 3.3 and Proposition 3.4 are wrong, see \textit{H. Rindler} [Proc. Am. Math. Soc. 114, No. 2, 561-563 (1992)]. Essentially inspired by Herfort, we prove that \(1\prec\pi_ G\) for certain tall prosolvable groups considered by \textit{W. Herfort} and \textit{P. Ribenboim} [Arch. Math. 42, 1-15 (1984; Zbl 0544.20026)] which are topologically generated by two elements (\(1\not\prec\pi_ G\) implies that \(G\) is topologically finitely generated). There exist connected tall groups \(G\) (all such groups are topologically generated by two elements) such that \(1\prec\pi_ G\).
Mathematics Subject Classification ID: 43A77
Mathematics Subject Classification ID: 22C05
Mathematics Subject Classification ID: 22D10
zbMATH DE Number: 4216160
zbMATH Keywords: compact tall groups
zbMATH Keywords: irreducible representations
zbMATH Keywords: left regular representations
zbMATH Keywords: tall prosolvable groups
zbMATH Keywords: topologically finitely generated
zbMATH Keywords: connected tall groups
author: Harald Rindler
MaRDI profile type: Publication
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cites work: Almost Periodic Operators in VN(G)
cites work: Property T and asymptotically invariant sequences
cites work: Amenable semigroups
cites work: Zu einem von B. H. und H. Neumann gestellten Problem
cites work: Die Eulersche Funktion endlicher auflösbarer Gruppen
cites work: Q5562332
cites work: Q3338434
cites work: A remark on discontinuous translation invariant functionals on \(L_ p (G)\) for certain compact groups \(G\)
cites work: Localization of profinite groups
cites work: Tall profinite groups
cites work: Varieties of groups
cites work: Q5550080
cites work: Groups of measure preserving transformations
cites work: Groups of measure preserving transformations. II
cites work: On Weak Containment Properties
cites work: Uniqueness of Invariant Means for Measure-Preserving Transformations
cites work: Translation-Invariant Linear Forms on L p (G)
cites work: Amenability, Kazhdan's property T, strong ergodicity and invariant means for ergodic group-actions
cites work: For 𝑛>3 there is only one finitely additive rotationally invariant measure on the 𝑛-sphere defined on all Lebesgue measurable subsets
description: scientific article; zbMATH DE number 4216160