20 JONATHAN ROSENBERG AND CLAUDE SCHOCHET

continuous-trace algebras with spectrum IRn are stably

commutative), we may assume

A * W®Co(f*n) (*)

where W is a stable continuous-trace algebra with spectrum G/H.

Our difficulty is that we don't know yet that the isomorphism (*)

can be made equivariant. For that we need the following

proposition.

PROPOSITION 2.9. Let G be a compact Lie group, not necessarily

connected, and let (A,a) be a stable continuous-trace G-algebra

with G-spectrum (G/H)XfR . Then there is a stable continuous-trace

G-algebra (W,o) with G-spectrum G/H and an exterior equivalence

(A,a) *

(W®Co(fRn)

,a®id) .

PROOF: Let W be the quotient of A corresponding to the closed

G-invariant subset (G/H)X{0) C

(G/H)XlRn.

Then W is a G-algebra

and A =W$C (f R ) , though not necessarily equivariantly. Let u be

the G-action on W. We want to compare a with u#id. The cocycle

g

"

ag (w g®id)

takes values in the inner automorphisms of A, by [RR, Theorem

0.8] if G is connected, but actually in general via [RR, Theorem

0.5(b)] since the map

H2((G/H)X{0},Z) • H2((G/H)XfRn,Z)

is always an isomorphism.

By [RR, Theorem 0.4], there is only one obstruction to

exterior equivalence of a with a$id, and it lies in the Moore

cohomology group HM(G,C((G/H)XIR , T)). Now we argue as in [Ro3,

Theorem 3.9]. Let C(X,T) denote the connected component of the

identity in C(X,T). The commuting diagram