Unitary representations and compact groups
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 \(1\prec\pi_ G\).
scientific article; zbMATH DE number 4216160
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