Groupoid cocycles and K-theory

TL;DR

This paper explores how cocycles on groupoids induce unbounded bimodules and index maps in K-theory, linking groupoid cocycle data to operator algebra invariants and index theory.

Contribution

It establishes a construction of unbounded bimodules from groupoid cocycles under mild conditions, extending the connection between groupoid dynamics and K-theoretic index maps.

Findings

Cocycles induce unbounded odd R-equivariant bimodules for groupoid C*-algebras.

The bimodule construction leads to index maps in K-theory when cocycles come from quasi-invariant measures.

The framework applies to integer-valued cocycles on étale groupoids, broadening the scope of index theory in groupoid C*-algebras.

Abstract

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