The completion theorem in K-theory for proper actions of a discrete group

We prove a version of the Atiyah-Segal completion theorem for proper actions of an infinite discrete group G. More precisely, for any finite proper G-CW-complex X, K*(EG x X-G) is the completion of K-G*(X) with respect to a certain ideal. We also show, for such G and X, that K-G(X) can be defined as the Grothendieck group of the monoid of G-vector bundles over X. (C) 2001 Elsevier Science Ltd. All rights reserved.