The E-theoretic descent functor for groupoids

Abstract

The paper establishes, for a wide class of locally compact groupoids $\Gamma$, the E-theoretic descent functor at the $C^{*}$-algebra level, in a way parallel to that established for locally compact groups by Guentner, Higson and Trout. The second section shows that $\Gamma$-actions on a $C_{0}(X)$-algebra $B$, where $X$ is the unit space of $\Gamma$, can be usefully formulated in terms of an action on the associated bundle $B^{\sharp}$. The third section shows that the functor $B\to C^{*}(\Gamma,B)$ is continuous and exact, and uses the disintegration theory of J. Renault. The last section establishes the existence of the descent functor under a very mild condition on $\Gamma$, the main technical difficulty involved being that of finding a $\Gamma$-algebra that plays the role of C_{b}(T,B)^{cont}$ in the group case.