Amenability and vanishing of \(L^2\)-Betti numbers: an operator algebraic approach
type: Item
instance of: scholarly article
title: Amenability and vanishing of \(L^2\)-Betti numbers: an operator algebraic approach
zbMATH Open document ID: 1279.46044
author: Vadim Alekseev
author: David Kyed
published in: Journal of Functional Analysis
publication date: 2012-08-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1105.3406
reviewed by: Ralf Meyer
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 46L89
Mathematics Subject Classification ID: 16E30
zbMATH DE Number: 6071596
zbMATH Keywords: \(L^2\)-Betti number
zbMATH Keywords: amenability
zbMATH Keywords: Følner condition
zbMATH Keywords: von Neumann dimension
MaRDI profile type: Publication
OpenAlex ID: W2008538261
arXiv ID: 1105.3406
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DOI: 10.1016/J.JFA.2012.05.016
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review text: This article unifies several previous results about vanishing of \(L^2\)-Betti numbers for amenable groups, quantum groups, and groupoids in the context of the definition of \(L^2\)-Betti numbers for von Neumann algebras by \textit{A. Connes} and \textit{D. Shlyakhtenko} [J. Reine Angew. Math. 586, 125--168 (2005; Zbl 1083.46034)]. The setup consists of a tower of algebras \(N\subseteq A\subseteq M\), where \(N\) and \(M\) are finite von Neumann algebras, \(A\) is a dense subalgebra in \(M\), and the inclusion \(N\to M\) is trace-preserving. The case \(N=\mathbb C\) is already interesting.NEWLINENEWLINEThe strong Følner condition introduced here is analogous to the Følner condition for groups. If \(N=\mathbb C\), \(A=\mathbb C[G]\) and \(N = LG\) for a discrete group \(G\), then the strong Følner condition holds if and only if the group \(G\) is amenable. A similar remark holds for discrete quantum groups.NEWLINENEWLINEIt is shown that the inclusion \(A\to M\) is dimension-flat, that is, the von Neumann dimension vanishes for the Tor-modules that are required to vanish for a flat ring inclusion. Furthermore, the strong Følner condition implies that~\(M\) is amenable relative to~\(N\); this is the usual notion of amenability or injectivity if \(N=\mathbb C\).NEWLINENEWLINEThe dimension-flatness implies that \(L^2\)-Betti numbers vanish for amenable groups, quantum groups, and measured groupoids or equivalence relations. Furthermore, \(L^2\)-Betti numbers of Connes-Shlyakhtenko vanish for irrational rotation algebras and UHF-algebras.
description: scientific article; zbMATH DE number 6071596