  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.
