18 JONATHAN ROSENBERG AND CLAUDE SCHOCHET

may be modified by spectral calculus to obtain an element z € A

with rf(z) a rank-one projection for K near K . As in [Di,4.4.4],

z defines a non-trivial continuous-trace ideal of A, and since z

is G-invariant, so is this ideal. This proves the theorem in the

case G = T.

Next we consider the case G a compact Lie group. Let G

denote the connected component of the identity, and consider its

action on A. We may assume that A is liminary, as before. By the

previous case, and reasoning as in [Di, 4.4.5], we see that for

every closed subgroup H of G with HST, A contains a dense open

H-invariant Hausdorff subset. But G contains a finite number of

o

one-parameter subgroups H1,...,H , each isomorphic to T, which

generate G algebraically (no closures needed). So choosing such

a dense open H.-invariant Hausdorff subset U. of A for each i and

letting U = U„A.. . fl U , we obtain a dense G -invariant open

1 x o

Hausdorff subset of A. (Density of U follows from the fact that

A is a Baire space.)

Choose representatives g. for the (finitely many) cosets of

G in G, and let Y = Ag.U. Again by the Baire property, Y is a

dense open G-invariant Hausdorff subset of A and corresponds to

some non-trivial G-invariant ideal J. Let C be some

continuous-trace ideal in J and let D be its G-saturation. Then D

is a non-trivial G-invariant ideal with Hausdorff spectrum and D

has local rank-one projections, thus D is continuous-trace. This

completes the proof of the theorem when G is a compact Lie group.

•

THEOREM 2.8. Let G be a compact Lie group, not necessarily

connected. If F is a collection of G-algebras and if each

continuous-trace G-algebra A with A = G/H (H running over closed

subgroups of G, G acting by translation) may be constructed from

F as described above, then F is a CQ-fundamental family.

Similarly, if F is the collection of commutative G-algebras of

the form C(G/H), then F is an AG-fundamental family.

PROOF: We shall prove only the statement about C -fundamental

families. The corresponding statement about Ap-fundamental

families is much easier, and uses (a proper subset of) exactly

the same arguments.