The theory is applied to prove the Hodge-De Rham-theorem for $L^2$-cohomology of oriented $boundary$-manifolds of bounded geometry. Another application are Bochner-type vanishing results for $L^2$-cohomology of compact and non-compact $boundary$-manifolds with certain curvature restrictions.
In the second part I specialize to coverings of compact $boundary$-manifolds. Suppose $(tilde A,tilde T)$ is the lift of an elliptic differential boundary value problem $(A,T)$. In this situation, the theory of von Neumann algebras provides us with a real valued dimension function, and I prove the following $L^2$-index theorem: the von Neumann index $ind_Gamma(tilde A,tilde T)$ is defined and equals the index of $(A,T)$. In the course of the proof I obtain the following result of general interest: Every smoothing $Gamma$-invariant operator on $tilde M$ is of von Neumann trace class. This would have been useful for example in the papers of Roe and Ramachandran.
To prove the theorem, we develop a theory of von Neumann traces for arbitrary Hilbert A-module morphisms. Here A is a von Neumann algebra with finite faithful normal trace.