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{U r f){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.) 5

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