This paper, http://arxiv.org/abs/1210.6100, has been accepted by the Proceedings of the Edinburgh Mathematical Society. I just sent off the copyright transfer form this evening, so everything is now set, I hope.
The paper is mostly paying an expository debt. In my CBMS lecture notes I said that if one has the Dirac operator on a complete spin manifold \(M\), and if there is some subset \(N\subseteq M\) such that \(D\) has uniformly positive scalar curvature outside \(N\), then the index of \(D\) belongs to the K-theory of the ideal \(I_N \triangleleft C^*(M) \) associated to the subset \(N\). A very special case of this is the observation of Gromov-Lawson that \(D\) is Fredholm if we have uniformly positive scalar curvature outside a compact set. There are of course analogous results using thepositivity of the Weitzenbock curvature term for other generalized Dirac operators.
Until now, I had not written up the proof of this assertion, but I felt last year that it was (past) time to do so. This paper contains the proof and also that of the associated general form of the Gromov-Lawson relative index theorem which also appears in my CBMS notes. The latter proof uses some results from my paper with Paul Siegel on sheaf theory and Pashcke duality.
The submission to PEMS is in honor of a very pleasant sabbatical spent in Edinburgh in fall 2004.