Groupoid cocycles and K-theory

Details Groupoid cocycles and K-theory Groupoid cocycles and K-theory

2010-05-20

arxiv.org/abs/1005.3677v4

Mathematics

K-Theory and Homology

Update: proof of exactness of integral cocycles corrected

Scientific paper

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 $K_{1}^{\R}(C^{*}(\mathcal{G}))\to \C$.

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