Property \(T\) for discrete groups in terms of their regular representation
type: Item
instance of: scholarly article
title: Property \(T\) for discrete groups in terms of their regular representation
zbMATH Open document ID: 0787.22007
DOI: 10.1007/BF01459515
author: Paul Jolissaint
published in: Mathematische Annalen
publication date: 1994-04-12T00:00:00Z
full work available at URL: https://eudml.org/doc/165145
review text: The main result of the article is a generalization of a theorem of A. Connes and V. Jones on Kazhdan's property \(T\) for discrete groups: it is proved (without any assumption on conjugacy classes) that a group \(\Gamma\) has property \(T\) if and only if the identity correspondence \(L^ 2(M)\) of the von Neumann algebra \(M\) of \(\Gamma\) admits a neighbourhood \(U\) such that any correspondence belonging to \(U\) contains some non zero subcorrespondence of \(L^ 2(M)\). Moreover, a new ideal \(\text{Bin}(\Gamma)\) of the Fourier-Stieltjes algebra \(B(\Gamma\times\Gamma)\) is introduced: It is the set of elements \(\varphi\) of \(B(\Gamma\times\Gamma)\) such that the one variable functions \(\varphi(\cdot,\gamma)\) and \(\varphi(\beta,\cdot)\) belong to the Fourier algebra \(A(\Gamma)\) for all fixed \(\beta\) and \(\gamma\). It is proved that every element of \(\text{Bin}(\Gamma)\) comes from a coefficient of some correspondence of \(M\), and property \(T\) of \(\Gamma\) is also expressed in terms of positive definite functions belonging to \(\text{Bin}(\Gamma)\).
Mathematics Subject Classification ID: 22D25
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 43A35
Mathematics Subject Classification ID: 43A65
zbMATH DE Number: 537329
zbMATH Keywords: Kazhdan's property \(T\)
zbMATH Keywords: discrete groups
zbMATH Keywords: von Neumann algebra
zbMATH Keywords: Fourier-Stieltjes algebra
zbMATH Keywords: positive definite functions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 537329