8 REPRESENTATION THEORY AND NUMERICAL AF-INVARIANTS

for the endomorphism

Ty—U(Toa)U*.

Cocycle equivalence of functions with values in groups G of unitaries have been

studied recently in ergodic theory; see, e.g., [73, 74]. Equation (1.33) above in

that setup is the assertion that U and U (taking values in the corresponding G)

are cohomologous.

Proof of Theorem 1.2. We will first verify that the relations (1.17)—(1.25) define a

representation of Od, and verify that its restriction to the abelian subalgebra

(1-37) Vd = C*\ sas*a

oo

a

l

e]Jzd

is the spectral representation. If g e L1 (fi, d/x), we have

(1.38) / g (x) dfi{x) = ^2 f g (x) dp (x)

dp(y)

dp(y) = *£[ 9(°i(v))

iJn d^{(rai{y))

= T2 9(°i (y)) p (^i

(y))~2

dn (y)

i

Jn

= j

a

( E 9(x)G(x)\ d»(y),

where G (x) = p(x) . (If it happens that *}2X. a(x)=yG (x) = 1, the relation

(1.38) says that p is cr-invariant, and p is then what is called a G-measure in [56].)

Applying (1.38) to g (x) = f (x, a (x)) we obtain

1.39) / / (x, a (x)) dp (x) = J2 [ f (^ (2/) v) P & (2/))"'

d»

M •

Jn

•

Jn

Defining Si by (1.19) and (1.21), we see immediately from the \% ix) term that the

ranges of Si are mutually orthogonal, and if £ G H, then from (1.39):

(1.40) ||S^||2= f

Xi

(x)p(x)2U(cr(x))\\2d^(x)

Jn

p(x)2U(a(x))fdn(x)

J

L

p(vi(y))2M(y)\\2p(vi(y)r2d»(y)

n

= iieir2

so each Si is an isometry, and hence

(1-41) S^Sj=6ijl.