L2-index theorem for manifolds with boundary
  Suppose M is a compact manifold with boundary. Let N be a normal covering of M. Suppose (A,T) is an elliptic differential boundary value problem on M with lift (\tilde A,\tilde T) to N. Then the von Neumann dimension of kernel and cokernel of this lift are defined. The main result of this paper is: these numbers are finite, and their difference, by definition the von Neumann index, equals the index of (A,T).   In this way, we extend the classical L^2-index theorem of Atiyah to manifolds with boundary. 
Thomas Schick