L 2 -invariants and Rank Metric

We introduce a notion of rank completion for bi-modules over a finite tracial von Neumann algebra. We show that the functor of rank completion is exact and that the category of complete modules is abelian with enough projective objects. This leads to interesting computations in the L 2-homology for tracial algebras. As an application, we also give a new proof of a Theorem of Gaboriau on invariance of L 2-Betti numbers under orbit equivalence. 1.1. Introduction. The aim of this article is to unify approaches to several results in the theory of L 2-invariants of groups, see [Lüc02, Gab02a], and tracial algebras, see [CS05]. The new approach allows us to sharpen several results that were obtained in [Tho06b]. We also give a new proof of D. Gaboriau's Theorem on invariance of L 2-Betti numbers under orbit equivalence. In order to do so, we introduce the concept of rank metric and rank completion of bi-modules over a von finite tracial von Neumann algebra. All von Neumann algebras in this article have a separable pre-dual. Recall, a von Neumann algebra is called finite and tracial, if it comes with a fixed positive, faithful and normal trace. Every finite (i.e. Dedekind finite) von Neumann algebra admits such a trace, but we assume that a choice of a trace is fixed. The rank is a natural measure of the size of the support of an element in a bi-module over a finite tracial von Neumann-algebra. The induced metric endows each bi-module with a topology, such that all bi-module maps are contractions. The main utility of completion with respect to the rank metric is revealed by the observation that the functor of rank completion is exact and that the category of complete modules is abelian with enough projective objects. Employing the process of rank completion, we aim to proof two main results. First of all, we will show that certain L 2-Betti number invariants of von Neumann algebras coincide with those for arbitrary weakly dense sub-C *-algebras. The particular case of the first L 2-Betti number was treated in [Tho06b]. The general result required a more conceptual approach and is carried out in this article. The importance of this result was pointed out to the author by D. Shlyakhtenko. Indeed, according to A. Connes and D. Shlyakhtenko, K-theoretic methods might be used to relate the L 2-Euler characteristic of a group C *-algebra to the …