Callias-type operators in C*-algebras

Abstract

A Dirac-type operator on a complete Riemannian manifold is of Callias-type if its square is a Schrödinger-type operator with a potential uniformly positive outside of a compact set. We present an index theorem for Callias-type operators twisted with Hilbert C*-module bundles. As an application, we derive an obstruction to the existence of Riemannian metrics of positive scalar curvature on noncompact spin manifolds in terms of closed submanifolds of codimension one. In particular, when N is a closed even dimensional spin manifold, we show that if the cylinder NxR carries a complete metric of positive scalar curvature, then the (complex) Rosenberg index on N must vanish.