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