EQUIVARIANT K-THEORY AND KK-THEORY 15
where B is the natural action of G on C*(K-L,a). D
REMARK 2.6. We note that C (K1,*) may be identified with the
induced algebra
IndHTGC*(KX/HX,a) * IndHtGEnd(V).
Since (H/K) carries a non-degenerate cocycle, it is canonically
self-dual, and V may also be viewed as the space of an
irreducible projective representation of (H/K).
Next we attempt to mimic the classical [Di, Ch. 4] argument
to produce a small fundamental family for G-algebras.
The following result is a necessary prerequisite to
understanding the structure of Type I G-algebras. For simplicity,
we restrict to the case of compact Lie groups, since this is the
only case we are interested in. The same result when G is any
compact, metrizable group could be deduced from [Ph, $8.1], which
gives a very different argument using only general topology.
However, our proof has the advantage of giving a more concrete
description of a G-invariant continuous-trace ideal in A, at
least when G = T.
THEOREM 2.7. Let G be a compact Lie group (not necessarily
connected) and let A be any Type I (separable) G-algebra. Then A
contains a non-zero G-invariant ideal of continuous-trace.
PROOF: We assume initially that G is a compact connected Lie
group. Without loss of generality, we may assume that A is
liminary, since the largest liminary ideal of A [Di, 4.2.6] is
invariant under all automorphisms of A, and in particular under
G. We now try as much as possible to imitate the proof of [Di,
Lemmes 4.4.2 and 4.4.4] in a G-eguivariant way. Using [Di, Lemme
4.4.3], choose a non-zero element y" of A+, say with ||y"| | ^ 1,
with Tr(*(y"))oo for all *€A. Let
= g*y"dg.
JG
Then y' has all the same properties as y" and it is also
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