A short proof of the approximation conjecture for amenable groups
type: Item
instance of: scholarly article
title: A short proof of the approximation conjecture for amenable groups
zbMATH Open document ID: 1158.28005
author: Daniel Pape
published in: Journal of Functional Analysis
publication date: 2008-12-02T00:00:00Z
review text: A standing conjecture, viz. Approximation Conjecture, in the theory of \(L^2\)-invariants, for \((R,G)\) where \(G\) is a discrete group has been proven in various cases and versions; especially the linearly ordered version of the conjecture for arbitrary group \(G\) and those directed systems \((G_i)_{i\in \mathbb{N}}\) which consist of finite index subgroups with integer coefficients \(R=\mathbb{Z}\) has been treated by numerous authors. In this paper, the author gives a short proof of Approximation Conjecture with complex coefficients for amenable groups; it is to be noted that a discrete group \(G\) is called amenable if there exists a linear functional \(m:l^\infty (G;R)\to R\) satisfying \(\inf\{\xi(g):g\in G\}\leq m(\xi)\leq \sup\{\xi(g):g\in G\}\).
Mathematics Subject Classification ID: 28D20
Mathematics Subject Classification ID: 22D40
zbMATH DE Number: 5376860
zbMATH Keywords: \(\ell ^{2}\)-Betti numbers
zbMATH Keywords: amenable groups
zbMATH Keywords: von Neumann dimension
zbMATH Keywords: approximation conjecture
reviewed by: Sobhakar Ganguly
MaRDI profile type: Publication
full work available at URL: j.jfa.2008.05.018
OpenAlex ID: W1968293368
cites work: Q3502037
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cites work: \(L^2\)-determinant class and approximation of \(L^2\)-Betti numbers
DOI: 10.1016/J.JFA.2008.05.018
description: scientific article; zbMATH DE number 5376860