Callias-Type Operators in von Neumann Algebras

We study differential operators on complete Riemannian manifolds which act on sections of a bundle of finite type modules over a von Neumann algebra with a trace. We prove a relative index and a Callias-type index theorem for von Neumann indexes of such operators. We apply these results to obtain a version of Atiyah's L-2-index theorem, which states that the index of a Callias-type operator on a non-compact manifold M is equal to the Gamma-index of its lift to a Galois cover of M. We also prove the cobordism invariance of the index of Callias-type operators. In particular, we give a new proof of the cobordism invariance of the von Neumann index of operators on compact manifolds.