The E-theoretic descent functor for groupoids

The paper establishes, for a wide class of locally compact groupoids $\mathrm{\Gamma }$, the E-theoretic descent functor at the $C^{\ast }$-algebra level, in a way parallel to that established for locally compact groups by Guentner, Higson and Trout. The second section shows that $\mathrm{\Gamma }$-actions on a $C_0(X)$-algebra $B$, where $X$ is the unit space of $\mathrm{\Gamma }$, can be usefully formulated in terms of an action on the associated bundle $B^{\mathrm{♯}}$. The third section shows that the functor $B\to C^{\ast }(\mathrm{\Gamma },B)$ 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 $\mathrm{\Gamma }$, the main technical difficulty involved being that of finding a $\mathrm{\Gamma }$-algebra that plays the role of C_{b}(T,B)^{cont}$ in the group case.