Chern characters for equivariant K -theory of proper G-CW-complexes by Wolfgang L¨ uck and Bob Oliver * In an earlier paper [10], we showed that for any discrete group G, equivariant K-theory for finite proper G-CW-complexes can be defined using equivariant vector bundles. This was then used to prove a version of the Atiyah-Segal completion theorem in this situation. In this paper, we continue to restrict attention to actions of discrete groups, and begin by con- structing an appropriate classifying space which allows us to define K * G (X ) for an arbitrary proper G-complex X . We then construct rational-valued equivariant Chern characters for such spaces, and use them to prove some more general versions of completion theorems. In fact, we construct two different types of equivariant Chern character, both of which involve Bredon cohomology with coefficients in the system ( G/H 7→ R(H ) ) . The first, ch * X : K * G (X ) ------→ H * G (X ; Q ⊗ R(-)), is defined for arbitrary proper G-complexes. The second, a refinement of the first, is a homomorphism e ch * X : K * G (X ) ------→ Q ⊗ H * G (X ; R(-)), but defined only for finite dimensional proper G-complexes for which the isotropy subgroups on X have bounded order. When X is a finite proper G-complex (i.e., X/G is a finite CW-complex), then H * G (X ; R(-)) is finitely generated, and these two target groups are isomorphic. The second Chern character is important when proving the completion theo- rems. The idea for defining equivariant Chern characters with values in Bredon cohomology H * G (X ; Q ⊗ R(-)) was first due to Slomi´ nska [14]. A complex-valued Chern character was constructed earlier by Baum and Connes [5], using very different methods. The completion theorem of [10] is generalized in two ways. First, we prove it for real as well as complex K-theory. In addition, we prove it for families of subgroups in the sense of Jackowski [9]. This means that for each finite proper G-complex X and each family F of subgroups of G, K * G (E F (G) × X ) is shown to be isomorphic to a certain completion of K * G (X ). In particular, when F = {1}, then E F (G)= EG, and this becomes the usual completion theorem. The classifying spaces for equivariant K-theory are constructed here using Segal’s Γ- spaces. This seems to be the most convenient form of topological group completion in our * Partly supported by UMR 7953 of the CNRS 1