Hilbert modules and modules over finite von Neumann algebras and applications to \(L^ 2\)-invariants
type: Item
instance of: scholarly article
title: Hilbert modules and modules over finite von Neumann algebras and applications to \(L^ 2\)-invariants
zbMATH Open document ID: 0886.57028
author: Wolfgang Lück
published in: Mathematische Annalen
publication date: 1998-05-06T00:00:00Z
review text: Let \({\mathcal A}\) be a finite von Neumann algebra. We show that the category of finitely generated Hilbert \({\mathcal A}\)-modules is equivalent to the category of finitely generated projective \({\mathcal A}\)-modules and that the category of finitely presented \({\mathcal A}\)-modules is an abelian category. This enables us to translate functional analytic invariants such as \(L^2\)-Betti numbers and Novikov-Shubin invariants to an algebraic setting of finitely presented \({\mathcal A}\)-modules. This allows to apply standard homological algebra to questions about \(L^2\)-invariants. We define refined \(L^2\)-invariants using the center-valued trace of \({\mathcal A}\). We discuss applications. For instance we show for an extension \(1\to \Delta\to \Gamma\to \pi\to 1\) of groups such that \(\Delta\) is finitely generated and infinite, \(\Gamma\) is finitely presented and \(\pi\) contains \(\mathbb{Z}\) as subgroup that the deficiency of \(\Gamma\) is bounded by 1. We show that all \(L^2\)-Betti numbers of Thompson's group \(F\) are trivial. This is a necessary condition for a positive answer to the still open question whether \(F\) is amenable.
reviewed by: Wolfgang Lück
Mathematics Subject Classification ID: 57R99
Mathematics Subject Classification ID: 46L99
zbMATH DE Number: 1121706
zbMATH Keywords: amenable group
zbMATH Keywords: \(L^ 2\)-Betti numbers
zbMATH Keywords: Novikov-Shubin invariants
MaRDI profile type: Publication
full work available at URL: s002080050112
OpenAlex ID: W2054220444
DOI: 10.1007/S002080050112
description: scientific article; zbMATH DE number 1121706