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
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