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We give an explicit construction of such a trace, and we should mention here that we were very much 43 inspired by Takesaki's approach [16]. So assume that 0 is a faithful normal positive linear functional and that or is the associated modular action. We may assume the existence of a separating and cyclic vector w E X such that O(x) = (xw, w) for all x E M, and of a strongly continuous oneparameter group { at } tER of unitaries such that vt(x) = at x at and a w = w for all t E R. This follows easily from the G.

Finally (M ®a R)(p (9 1) is the von Neumann algebra on p3C 0 L2(R) generated by 7i(x)(p 0 1) = ir(xp) and X(s)(p®1) = p ® Xs. , pJC) is 1 OX s . Then the result follows. We will now consider the cases M semi-finite and M type III separately. This makes sense because of the previous proposition. If M is semi-finite it is well known that the modular automorphisms are inner [15]. In fact this could be deduced from Connes' cocycle Radon-Nikodym theorem for weights and the fact that the modular action associated to the trace is the trivial one.

23 If a is spatial, that is if there exists a continuous unitary representation a : s - as of G in 3C such that 3. 12 Theorem. a s(x) = as xa* then S (M ®a G)'= {x'®1, as®pSIX' EM', S EG)11 Proof. By theorem 3. 10 we have M ®a G = {xEM ®63(L2(G)) I6t(x) = x, Vt E G ). Now 0t(x) = x means (at ® pt)x(at ®pt)* = x as 0t = at ® ad pt. Hence M®aG=M®63(L2(G)) n {as®psIs EG}' and (M®a G)'=((M'®1) u {as®psls EG})". To finish this section, as promised we give a proof of proposition 2. 13 of the end of the previous section.

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A. F. Lavriks truncated equations by Kaufman R. M.

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