EQUIVARIANT K-THEORY AND KK-THEORY 15

where B is the natural action of G on C*(K-L,a). D

REMARK 2.6. We note that C (K1,*) may be identified with the

induced algebra

IndHTGC*(KX/HX,a) * IndHtGEnd(V).

Since (H/K) carries a non-degenerate cocycle, it is canonically

self-dual, and V may also be viewed as the space of an

irreducible projective representation of (H/K).

Next we attempt to mimic the classical [Di, Ch. 4] argument

to produce a small fundamental family for G-algebras.

The following result is a necessary prerequisite to

understanding the structure of Type I G-algebras. For simplicity,

we restrict to the case of compact Lie groups, since this is the

only case we are interested in. The same result when G is any

compact, metrizable group could be deduced from [Ph, $8.1], which

gives a very different argument using only general topology.

However, our proof has the advantage of giving a more concrete

description of a G-invariant continuous-trace ideal in A, at

least when G = T.

THEOREM 2.7. Let G be a compact Lie group (not necessarily

connected) and let A be any Type I (separable) G-algebra. Then A

contains a non-zero G-invariant ideal of continuous-trace.

PROOF: We assume initially that G is a compact connected Lie

group. Without loss of generality, we may assume that A is

liminary, since the largest liminary ideal of A [Di, 4.2.6] is

invariant under all automorphisms of A, and in particular under

G. We now try as much as possible to imitate the proof of [Di,

Lemmes 4.4.2 and 4.4.4] in a G-eguivariant way. Using [Di, Lemme

4.4.3], choose a non-zero element y" of A+, say with ||y"| | ^ 1,

with Tr(*(y"))oo for all *€A. Let

y« = g*y"dg.

JG

Then y' has all the same properties as y" and it is also