<em>L</em>²-Betti Numbers of Infinite Configuration Spaces

Abstract

The space Γ_X_ of all locally finite configurations in a infinite covering X of a compact Riemannian manifold is considered. The de Rham complex of square-integrable differential forms over Γ_X_, equipped with the Poisson measure, and the corresponding de Rham cohomology and the spaces of harmonic forms are studied. A natural von Neumann algebra containing the projection onto the space of harmonic forms is constructed. Explicit formulae for the corresponding trace are obtained. A regularized index of the Dirac operator associated with the de Rham differential on the configuration space of an infinite covering is considered.