EQUIVARIANT K-THEORY AND KK-THEORY 19

Let A be a Type I G-algebra. By repeated use of Theorem 2.8,

we see that A has a composition series {A } where A = 0,

AwJ./Aw is a non-zero G-invariant ideal of A/Aa with

a+l a

continuous-trace, and

A„ = lim {A,:*a}

CI • * JB

for a a limit ordinal. Since we are assuming that A is separable,

we have Aw * A for some countable ordinal a. So it suffices to

a

show that any continuous-trace G-algebra is generated by F.

The next step is the reduction to the case of one orbit

type. Suppose that A is a continuous-trace G-algebra with

spectrum X. Then X contains an open subset with one orbit type,

by [MZ, p. 222]. We divide by the corresponding G-invariant ideal

and repeat the argument on the quotient algebra. By a limit

argument we can reduce to the case of a finite number of orbit

types. By iterated extensions, we can reduce to the case of a

single orbit type. So suppose that X has a single orbit type- say

ti

all stability groups are conjugate to H. Then X is a free

NQ(H)/H -space and

X a GXNr(H)xH-

G

JJ

By Gleason's cross-section theorem [MZ, pp. 219-221], X has a

covering by open sets U , NQ(H)-isomorphic to (NQ(H)/H)XS for

certain locally compact S . Taking the induced cover of X, then

extracting a countable subcovering, we see from passage to limits

that we may assume that X = (G/H)XS, S locally compact. Taking

one-point compactifications, we may assume without loss of

generality that S is compact. Then S is a projective limit of

finite simplicial complexes, and so X is a limit of finite

G-complexes (e.g., in the sense of [May]) built out of G-cells of

the form (G/H)xfRn. So we have reduced to the case where A is a

continuous-trace algebra with A = (G/H)XIR , where G acts

transitively on G/H and trivially on IRn.

Now the problem is to classify continuous-trace G-algebras

with G-spectrum (G/H)xiRn. Taking the algebra to be G-stable and

using the fact that lRn is contractible (and hence that all