Spectral flow for nonunital spectral triples
  We prove two results about nonunital index theory left open by [CGRS2]. The first is that the spectral triple arising from an action of the reals on a C*-algebra with invariant trace satisfies the hypotheses of the nonunital local index formula. The second result concerns the meaning of spectral flow in the nonunital case. For the special case of paths arising from the odd index pairing for smooth spectral triples in the nonunital setting we are able to connect with earlier approaches to the analytic definition of spectral flow. 
A. Carey, V. Gayral, J. Phillips, A. Rennie, F. Sukochev
math.KT math.FA