EQUIVARIANT K-THEORY AND KK-THEORY 17
topology. The fact that y is G-invariant implies that 7(g)
commutes with v.(g) for h€H. Since H is a finite cyclic group, it
is easy to choose p(l) to be a rank-one subprojection of y(l)
commuting with {v.(l):h€H}. Then
p(t) = v _ (l)_1p(l)v _ (1) t€G
t t
gives a well-defined rank-one projection p C(G/H,JT), since
given t€G/H, v (1) is well-defined modulo unitaries commuting
t
with p(l). From the cocycle identity for v we compute that
v.(g)p(t-1g) = v.{g)v (l)_1p(l)v (1)
9 t g
1t
" v _i(l)"lv 1 (1)V 1 (l)_1P(l)v _ (1)
g g" t g xt g 1t
= v (i)
1pCi)v
(i)
g g t
v _1(l) Ap(l)v ^(Dlv ^(1) v _1 (1))
g g g g t
- P(t)vt(g),
which shows that p is G-invariant.
Having produced a G-invariant rank-one projection p in B, we
let x A be any element projecting to p. Averaging x under G,
we may assume that x is G-invariant, and also we may assume that
1/2xy1/2
||x| | = 1. Then y is G-invariant and dominated by y, so the
function
1/2 1/2
K ~ Tr(*(?'' xy"))
is continuous at K , by [Di, 4.4.2(1)], and of course
G-invariant. Since
*0Y1/2xy1/2)
= *0(YPY) = *Q(P)
1/2xy1/2
is of rank one, we conclude as in [Di, 4.4.2(H)] that y
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