The E-theoretic descent functor for groupoids

The paper establishes, for a wide class of locally compact groupoids Γ , 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. Section 2 shows that Γ -actions on a C0(X)-algebra B, where X is the unit space of Γ , can be usefully formulated in terms of an action on the associated bundle B . Section 3 shows that the functor B → C∗(Γ,B) is continuous and exact, and uses the disintegration theory of J. Renault. Section 4 establishes the existence of the descent functor under a very mild condition on Γ , the main technical difficulty involved being that of finding a Γ -algebra that plays the role of Cb(T ,B)cont in the group case.