Groupoid cocycles and K-theory

Abstract

Let $c:\mathcal{G}\to \mathbb{R}$ 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 étale groupoids), $c$ gives rise to an unbounded odd $\mathbb{R}$-equivariant bimodule $(\mathpzc{E},D)$ for the pair of $C^{\ast }$-algebras $(C^{\ast }(\mathcal{G}),C^{\ast }(\mathcal{H}))$. If the cocycle comes from a continuous quasi-invariant measure on the unit space ${\mathcal{G}}^{(0)}$, the corresponding element in $K{K}_1^{\mathbb{R}}(C^{\ast }(\mathcal{G}),C^{\ast }(\mathcal{H}))$ gives rise to an index map $K_{1}^{\R}(C^{*}(\mathcal{G}))\to \C$.

Update: proof of exactness of integral cocycles corrected

Cited by in corpus (1)

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