CHAPTER 4 v CONSTRUCTION OF M NOTATION. Let for every a! G M', r G E, / G H: ( J V ) / ) ^ y) = c ( ^ 2/ » yW(y)c(x, y, y)~xf{x, y) c(z, T ~ V T~ly)-lf(x, r _ 1 y) Then I'(a'),n'(T) G # ( # ) . Moreover, 7r'(r) is unitary. NOTATION. Let M v be the von Neumann algebra generated by I'(af) and 7r(r) for all a' G M'', r G E. Theorem 4.1 Every operator in M v commutes with every operator in M. PROOF. It is enough to check that for a G M, a' G M', r, cr G E: /(a) and 7r(r) commute with I'(a') and 7r'(cr). It can be checked by the straight calculation. 6
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