Institute of Mathematics of NAS of Ukraine,
3 Tereshchenkivs'ka Str.,
01601 Kyiv-4, UKRAINE

L²-Betti numbers of Poisson configuration spaces

Abstract:
The space G_X of all locally finite configurations in a infinite covering X of a compact Riemannian manifold is considered. The deRham complex of square-integrable differential forms over G_X, equipped with the Poisson measure, and the corresponding deRham cohomology and the spaces of harmonic forms are studied. We construct a natural von Neumann algebra which contains the projection onto the space of harmonic forms and obtain explicit formulae for the corresponding trace.