The E-theoretic descent functor for groupoids

Abstract

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 C 0 ( 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 C b ( T , B ) cont in the group case.