1. GENERAL REPRESENTATIONS OF Od ON A SEPARABLE HILBERT SPACE 11
we have by (1.55)
(1-59) TiaTia = SiaSia
TTrPrP rji*rji*Tj*
= UT T* U*
and hence U commutes with the representatives on H of the algebra XV Hence U
has a decomposition
(1.60) U= U(x) dfi(x)
Jn
where Q 3 x ^ U (x) is a measurable field on unitaries. It now follows from (1.54)
and (1.57) that Si has the form (1.21). This ends the proof of Theorem 1.2.
Proof of Theorem 1.4. Adopt the assumptions in Theorem 1.4 and let T be an
intertwiner between the two representations. In particular this means that T inter-
twines the two spectral representations of Vj on H and 7Y, respectively, i.e.,
(1.61) TSaS*a = Sj*T
for all multi-indices a. But this is equivalent to Q having the decomposition (1.30)
and T having the measurable decomposition
/•e
(1.62) T= T(x) dn(x)
r&
where T (x) £ B (H (x), H (x)). We now compute, using (1.21),
(1.63) TSiZ(x)=T(x)(Sit)(x)
= Xi(x)p(x)T(x)U{x)£(a(x))
and
(1.64) StTZ (x) =
Xi
{x) p (x) U (x) (TC) (a (*))
= Xi(x)p(x)U{x)T(cr{x))S(a{x)).
Using the intertwining property (1.31) we thus deduce that
(1.65) T (x) U(x) = U (x) T (a (x)).
Conversely, if T satisfies (1.65), the intertwining follows from (1.63) and (1.64).
This ends the proof of Theorem 1.4.
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