The E-theoretic descent functor for groupoids

摘要

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. Section 2 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-#. Section 3 shows that the functor B -> C* (Gamma, 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 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. (c) 2008 Elsevier Inc. All rights reserved.