Dimension theory of arbitrary modules over finite von Neumann algebras and \(L^2\)-Betti numbers. II: Applications to Grothendieck groups, \(L^2\)-Euler characteristics and Burnside groups. (English) Zbl 1001.55019
Summary: We continue the study [Part I: ibid. 495, 135-162 (1998; Zbl 0921.55016)] of the generalized dimension function. We detect elements in the Grothendieck group \(G_0(\mathbb{C}\Gamma)\) of finitely generated \(\mathbb{C}\Gamma\)-modules, 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 \(\Gamma\).
MSC:
| 58J22 | Exotic index theories on manifolds |
| 19K33 | Ext and \(K\)-homology |
| 46L85 | Noncommutative topology |
