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
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