L^2-Invariants and rank metric

Details L^2-Invariants and rank metric L^2-Invariants and rank metric

2006-07-11

arxiv.org/abs/math/0607263v1

Mathematics

Operator Algebras

10 pages, no figures

Scientific paper

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.

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