CHAPTER 3
UNITARY REALIZATION OF a(yx)
Let fa e H be a, separating and cyclic vector for M, 0o(#,?/) = ^(x^^oiv)-
NOTATION.
Let for r G E, a G M: aT(a)(x) =
a^x^T-ix^(a(r~1x)).
Then x i— aT(a)(x) is measurable, i. e. there exists
aT(a) / ar(a)(x)dfi(x) G M
Jx
It is clear that aT is a *-automorphism of M. It is not hard to see that aTl o aT2 =
aTlT2 for (ri,r
2
G E).
Let du~1(x^y) = dv(y,x) and D(x,y) = [dv"1 /dv)(x,y). It is known (see [14,
Corollary 2 on page 294]) that for a. e. # ~ y ~ z: -D(x, y)D(x, z) D(x, z).
By the general theorem [6, Corollary 2.5.32], for r G E there exist unitary
operators UT in B(iT) such that aT(a) = UTaU*, UTlUT2 = Z7riT2 (hence C/T-i = £/*)
and UTJ JUT, where J is the unitary involution from the Tomita-Takesaki theory,
associated with M and 0o-
By [14, Proposition 2.2], for 0 G E we have: (d(f)^fi/dfi)(y) =
D((j~1y,y)
for
a. e. x G X. Thus,
dfj,((j)~1y)
= d(f)*/j,(y) =
D((f)~ly,y)dii{y).
Take 0 = r
_ 1
and get
d^(Ty) = D(ry,y)dfi(y)
Moreover, for a e A: UTaU* = a
r
(a) = a o r
_ 1
. By [16, Proposition 4.1], this
implies that there exist unitary operators Ux from H(x) onto H{rx) defined a. e.
such that (UTf)(x) =
D[X,T-XXYXI2VTT_XJ[T-XX)
for a. e. x G X, / G if.
NOTATION.
Let t/(a;,T-ix) = ^ - i ^ -
The notation above is well defined for a. e. x G X, i. e. it depends only on r _ 1 x ,
not on r G E.
REMARK.
By the definition, for / G H, a. e. :r G X:
CW-i*)/^-
1
*) =
D{x,r-lx)^2{Urf){x)
It can be checked immediately that for a. e. x G X, y ~ z ~ x, a G M(x):
1- l/(y,x) = U{y,z)U(z,x);
2. a(3/,x)(a) = U(y,x)aU(x,y)]
3. J{y)U(ViX) =Uiy,x)J(x).
Corollary 3.1 o:(x3/)(J(2/)) = J(x) a. e.
(See the Remark on a(x,y) at the end of Section 2.)
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