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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