Dimension theory of arbitrary modules over finite von Neumann algebras and applications to $L^2$-Betti numbers

Abstract

We define for arbitrary modules over a finite von Neumann algebra $\cala$ a dimension taking values in $[0,\infty]$ which extends the classical notion of von Neumann dimension for finitely generated projective $\cala$-modules and inherits all its useful properties such as additivity, cofinality and continuity. This allows to define $L^2$-Betti numbers for arbitrary topological spaces with an action of a discrete group $\Gamma$ extending the well-known definition for regular coverings of compact manifolds. We show for an amenable group $\Gamma$ that the $p$-th $L^2$-Betti number depends only on the $\cc\Gamma$-module given by the $p$-th singular homology. Using the generalized dimension function we detect elements in $G_0(\cc\Gamma)$, provided that $\Gamma$ is amenable. We investigate the class of groups for which the zero-th and first $L^2$-Betti numbers resp. all $L^2$-Betti numbers vanish. We study $L^2$-Euler characteristics and introduce for a discrete group $\Gamma$ its Burnside group extending the classical notions of Burnside ring and Burnside ring congruences for finite $\Gamma$. Keywords: Dimension functions for finite von Neumann algebras, $L^2$-Betti numbers, amenable groups, Grothendieck groups, Burnside groups