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/))"'

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.
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