1. for every (x,y) G R, ct(x^ is an isomorphism from M(y) onto M(x);
2. for x ~ y ~ z: c*(x,z) = ot(x,y) °
3. if a = Jx a(x)d/j,(x) G M, r G E, then x i— ft(X)TI)(a(rx)) is a measurable
operator field (i. e. there exist b G M such that 6(x) =
a. e. ).
= {(x, y, z) | £, J/, 2 G X,
~ 2/ ~ *}.
Let c :
\JzeX B(H(z)) be a map with the following properties:
1. for every (x, y, z) G
c(x, ?/, z) is a unitary operator on H(z)\
2. for every (x,y,z) G i?2, c{x,y,z) normalises M(z), i. e.
= M(z)
3. for a. e. x G X and for every y~z~v~w~x:
G M(z)
c(x, y, z)c(y, v, z)c(x, v, z)~l = a ^ ) (c(x, y, w)c(?/, v, w)c(x, v, w)'1)
4. for a. e. x G X and for y ~ 2 ~ t; ~ x:
If 0o is a separating and cyclic vector for M, then for a. e. y G X,
(/o(y) is a separable and cyclic vector for M(y), so for a. e. (x,y) G R, ct(x,y)
is spatial, so it can be expanded to all of B(H(y)).
5. for every (x,y,z) G R2: c(x,x,y) = lH(y), and c(x,y, z)~l = c(y,x, z)\
6. for every r,a G E, the map x i-» c(rx,ax,x) is Borel, i. e. there exists
a fx B(H(x))d/jJ(x) such that a(x) = c(rx,ax,x) a. e.
For every a G M, r G E we define:
(I(a)f)(x,y) = a{y:X)(a(x))f(x,y), feH
(ir(r)f)(x, 2/) = c(x, rx, y)f(rx, y), feH
where a = Jx a(x)d/j,(x) is the decomposition of a.
Then because of the measurability of a and c: / ( a ) / G H, 7r(r)/ G # , and
further: 1(a) G B(H), 7r(r) G # ( # ) for every such a and r. Moreover, for every
r G E the operator TT(T) is unitary.
Definition 2.1 Tfee von Neumann algebra M generated by 1(a) and 7T(T) /or all
a G M, r G E will be called a crossed product of the von Neumann algebra M
by the equivalence relation R. We will write:
M = Af Ml?
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