Groupoid cocycles and K-theory

Let c : G → R be a cocycle on a locally compact Hausdorff groupoid G with Haar system, and H the subgroupoid ker c ⊂ G. Under some mild conditions (satisfied by e.g. all integral cocycles on an ´etale groupoid), c gives rise to an unbounded odd R-equivariant bimodule (E,D) for the pair of C -algebras (C (G),C (H)). If the cocycle comes from a continuous quasi-invariant measure on the unit space G(0), the corresponding element [(E,D)] in KK1(C (G),C (H)) gives rise to an index map K1(C (G)) → C.