Hilbert modules and modules over finite von Neumann algebras and applications to \(L^ 2\)-invariants
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.
scientific article; zbMATH DE number 1121706