X iv : d g - ga / 9 70 70 11 v 1 1 0 Ju l 1 99 7 “ Dimension theory of arbitrary modules over finite von Neumann algebras and applications to L 2 -

We define for arbitrary modules over a finite von Neumann algebra A a dimension taking values in [0,∞] which extends the classical notion of von Neumann dimension for finitely generated projective A-modules and inherits all its useful properties such as Additivity, Cofinality and Continuity. This allows to define L2-Betti numbers for arbitrary topological spaces with an action of a discrete group Γ extending the wellknown definition for regular coverings of compact manifolds. We show for an amenable group Γ that the p-th L2-Betti number depends only on the CΓ-module given by the p-th singular homology. Using the generalized dimension function we detect elements in G0(CΓ), provided that Γ is amenable. We investigate the class of groups for which the zero-th and first L2-Betti numbers resp. all L2-Betti numbers vanish. We study L2Euler characteristics and introduce for a discrete group Γ its Burnside group extending the classical notions of Burnside ring and Burnside ring congruences for finite Γ.