Toeplitz operators and the Roe-Higson type index theorem in Riemannian   surfaces
  We study a two dimensional analogue of the Roe-Higson index theorem for a partitioned manifold. We prove that Connes' pairing of some invertible element with Roe's cyclic one-cocycle coincides to the Fredholm index of a Toeplitz operator. In the proof of this paper, we use some properties of a circle and use Higson's argument. In the last section, there is a example of partitioned manifold, which is not a cylinder, with non-trivial pairing. 
Tatsuki Seto
math.DG