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,∞]
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
Γ
extending the well-known definition for regular coverings of compact manifolds. We show for an amenable group
Γ
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
Γ
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
Γ
its Burnside group extending the classical notions of Burnside ring and Burnside ring congruences for finite
Γ
.
Keywords: Dimension functions for finite von Neumann algebras,
L
2
-Betti numbers, amenable groups, Grothendieck groups, Burnside groups