L²-Invariants and rank metric

Original · EN

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 L²-homology for tracial algebras. As an application, we also give a new proof of a Theorem of Gaboriau on invariance of L²-Betti numbers under orbit equivalence.