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.
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Sergio Albeverio, Alexei Daletskii, <em>L</em><sup>2</sup>-Betti Numbers of Inﬁnite Conﬁguration Spaces. Publ. Res. Inst. Math. Sci. 42 (2006), no. 3, pp. 649–682DOI 10.2977/PRIMS/1166642153