The Atiyah-Segal completion theorem for C*-algebras

We generalize the Atiyah-Segal completion theorem to C^*-algebras as follows. Let A be a C^*-algebra with a continuous action of the compact Lie group G. If K_*^G(A) is finitely generated as an R(G)-module, or under other suitable restrictions, then the I(G)-adic completion K_*^G(A) is isomorphic to RK_*([A C(EG)]^G), where RK_* is representable K-theory for - C^*-algebras and EG is a classifying space for G. As a corollary, we show that if and are homotopic actions of G, and if K_*(C^*(G,A,)) and K_*(C^*(G,A,)) are finitely generated, then K_*(C^*(G,A,))K_*(C^*(G,A,)). We give examples to show that this isomorphism fails without the completions. However, we prove that this isomorphism does hold without the completions if the homotopy is required to be norm continuous.This work was partially supported by an NSF Graduate Fellowship and by an NSF Postdoctoral Fellowship.