4

IGOR FULMAN

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) °

a(y,z)\

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) =

ce(a:,rx)(«(/7"^))

a. e. ).

Let

R2

= {(x, y, z) | £, J/, 2 G X,

JC

~ 2/ ~ *}.

Let c :

R2

— \JzeX B(H(z)) be a map with the following properties:

1. for every (x, y, z) G

R2,

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.

c(x,y,z)M(2:)c(x,2/,2:)"1

= M(z)

3. for a. e. x G X and for every y~z~v~w~x:

c(x,y,z)c(y,v,z)c(x,v,z)~l

G M(z)

and

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:

REMARK.

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?