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.

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