Discrete groups with Kazhdan's property \(T\) and factorization property are residually finite
type: Item
instance of: scholarly article
title: Discrete groups with Kazhdan's property \(T\) and factorization property are residually finite
zbMATH Open document ID: 0805.46066
DOI: 10.1007/BF01459798
author: Eberhard Kirchberg
published in: Mathematische Annalen
publication date: 1994-11-09T00:00:00Z
full work available at URL: https://eudml.org/doc/165222
review text: Let \(G\) be a discrete group. \(G\) has factorization property (property (F)) if the two-sided regular representation \(\lambda\otimes \rho: G\times G\to {\mathcal L}(\ell_ 2(G))\) given by \(\lambda\otimes \rho(g,h)= \lambda(g)\rho(h)\) extends to a continuous *-representation from \(C^*(G)\otimes^{\min} C^*(G)\) into \({\mathcal L}(\ell_ 2(G))\) [cf. the author, Invent. Math. 112, 449-489 (1993), sec. 7]. Let \(G\) be a discrete group with property \(T\) of \textit{D. A. Kazhdan} [Funct. Anal. Appl. 1, 63-65 (1967; Zbl 0168.276)]. We show that \(G\) is maximally almost periodic if and only if \(G\) has property (F). Since for discrete groups with property \(T\), maximal almost periodicity is equivalent to residual finiteness, it follows that property (F) is not satisfied for all discrete groups. In particular, \(G\) is residually finite if it has property \(T\) and has a faithful representation into a locally compact group \(H\) such that \(C^*(H)\) is nuclear (e.g. if \(H\) is a connected Lie group) or if \(G\) has property \(T\) and a faithful unitary representation into the hyperfinite \(\text{II}_ 1\) factor.
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 22D25
Mathematics Subject Classification ID: 46L35
zbMATH DE Number: 683113
zbMATH Keywords: property (F)
zbMATH Keywords: discrete group with property \(T\) of Kazhdan
zbMATH Keywords: factorization property
zbMATH Keywords: two-sided regular representation
zbMATH Keywords: maximal almost periodicity
zbMATH Keywords: residual finiteness
zbMATH Keywords: nuclear
zbMATH Keywords: Lie group
zbMATH Keywords: faithful unitary representation
zbMATH Keywords: hyperfinite \(\text{II}_ 1\) factor
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 683113