Let $c:\mathcal{ G}\to\R$ be a cocycle on a locally compact Hausdorff groupoid $\mathcal{ G}$ with Haar system. Under some mild conditions (satisfied by all integer valued cocycles on \'{ e}tale groupoids), $c$ gives rise to an unbounded odd $\R$-equivariant bimodule $(\mathpzc{ E},D)$ for the pair of $C^{ *}$-algebras $(C^{ *}(\mathcal{ G}),C^{ *}(\mathcal{ H}))$. If the cocycle comes from a continuous quasi-invariant measure on the unit space $\mathcal{ G}^{ (0)}$, the corresponding element in $KK_{ 1}^{ \R}(C^{ *}(\mathcal{ G}),C^{ *}(\mathcal{ H}))$ gives rise to an index map $K_{ 1}^{ \R}(C^{ *}(\mathcal{ G}))\to \C$.
Categories: math.KT