EQUIVARIANT K-THEORY AND KK-THEORY 7

Most of the time a is injective and then K*(S) • 0 and K°(S) has

homological dimension 1. Otherwise a = 0 and KQ(S) is R(G)-free.

Here is another type of example. Let T be a torus, and let

us suppose that T is embedded as a maximal torus in a simply

connected compact Lie group G. Let H be any closed subgroup of G.

Then G/H is a T~space (one can get circle actions on

3-dimensional lens spaces, for instance), and

K*(G/H) 2 R(T)®R(G)K*(G/H) 2 R(T)*R(Q)R(H),

concentrated in degree 0 (though possibly with big homological

dimension).

We briefly mention one other consequence of our results: for

groups G satisfying Hodgkin's condition one obtains a fairly

definitive answer to a question raised in [Pa]. By a theorem of

Green [Gr2] and Julg [Jul], one knows that for any compact group

G and any G-algebra A, there is a natural isomorphism

K*(A) 2 K#(GKA)

*

where GKA denotes the C -crossed product. (The case A = C(X) and

G finite had actually been treated much earlier by Atiyah.)

Paulsen points out (and in fact this is done much more generally

in [Ka3]) that the analogous result for the dual theory holds if

G is finite, but not in general, and he raises the question of

* *

determining the precise relationship between KQ(A) and K (GKA).

We see in fact that for good connected groups, the relationship

is given by a spectral sequence

ExtjJ(Q)(K,,(GKA),R(G)) —» K*(A),

while we have a short exact sequence

0 —*Ext2(K#(GrA),2) •

K*(GKA)

• Hom2(K#

(GKA)

,Z) —+ 0.

which splits, unnaturally. It's no wonder one sees no obvious

connection between K (GKA) and K_(A).