A characterization of locally compact amenable groups by means of tensor products
type: Item
instance of: scholarly article
title: A characterization of locally compact amenable groups by means of tensor products
zbMATH Open document ID: 0672.22003
DOI: 10.1007/BF01198348
published in: Archiv der Mathematik
publication date: 1989-01-01T00:00:00Z
review text: There is a variety of characterizations of amenability of a locally compact group G. One, which is due to Hulanicki and Reiter and proved very important in representation theory, says that G is amenable if and only if the trivial l-dimensional representation \(l_ G\) is weakly contained in the left regular representation \(\lambda_ G\) \((l_ G\prec \lambda_ G)\). For unitary representations \(\pi\) and \(\rho\) of G, denote by \(\pi\) \(\otimes \rho\) their tensor product and by \({\bar \pi}\) the conjugate representation of \(\pi\). Observing that \(\lambda_ G\otimes {\bar \lambda}_ G\) is a multiple of \(\lambda_ G\), the Hulanicki- Reiter theorem implies that G is amenable provided that \(l_ G\prec \pi \otimes {\bar \pi}\) for some representation \(\pi\) such that \(\pi \prec \lambda_ G\). The author proves the following nice and important converse result: If G is amenable, then \(l_ G\prec \pi \otimes {\bar \pi}\) for any unitary representation \(\pi\) of G. This had previously been shown in a number of particular cases. A main tool in the proof is that \(\pi\) \(\otimes {\bar \pi}\) can be realized on the space of Hilbert- Schmidt operators in the Hilbert space of \(\pi\).
Mathematics Subject Classification ID: 22D10
Mathematics Subject Classification ID: 43A07
zbMATH DE Number: 4099569
zbMATH Keywords: trivial representation
zbMATH Keywords: weak containment
zbMATH Keywords: amenability
zbMATH Keywords: locally compact group
zbMATH Keywords: unitary representations
zbMATH Keywords: tensor product
zbMATH Keywords: Hulanicki-Reiter theorem
author: Bachir Bekka
reviewed by: Eberhard Kaniuth
MaRDI profile type: Publication
cites work: A Galois-correspondence for general locally compact groups
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full work available at URL: https://doi.org/10.1007/bf01198348
OpenAlex ID: W2094303634
description: scientific article; zbMATH DE number 4099569