Dimension theory of arbitrary modules over finite von Neumann algebras and applications to $L^2$-Betti numbers
| dc.creator | Lueck, Wolfgang | |
| dc.date | 1997-07-10 | |
| dc.date.accessioned | 2026-07-07T09:13:15Z | |
| dc.date.available | 2026-07-07T09:13:15Z | |
| dc.description | 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 $Γ$ 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 $\ccΓ$-module given by the $p$-th singular homology. Using the generalized dimension function we detect elements in $G_0(\ccΓ)$, 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 | |
| dc.description | 34 pages, AMS-Latex2e | |
| dc.identifier | https://arxiv.org/abs/dg-ga/9707011 | |
| dc.identifier | http://arxiv.org/abs/dg-ga/9707011 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/152260 | |
| dc.subject | Differential Geometry | |
| dc.subject | 55T99, 46L99 | |
| dc.title | Dimension theory of arbitrary modules over finite von Neumann algebras and applications to $L^2$-Betti numbers | |
| dc.type | text |