1. GENERAL REPRESENTATIONS OF Od ON A SEPARABLE HILBERT SPACE 7

and

(1.29) H= I H(x) ^0* 0

as defined in Theorem 1.2. Partition Q into three a-invariant Borel sets

(1.30) 0 = QoUQiU0

2

such that \i and jl are equivalent on Qo, p,(£li) = 0, and //(f^) = 0- Then an

operator T G B(H,H) is an intertwiner between the two representations, i.e.,

(1.31) TS% = SiT,

if and only if T has a measurable decomposition

(1.32) T = T(x) dfi(x)

where T (x) G B (H(x) ,H(x)) and

(1.33) T (x) U{x) = U (x) T (a (x))

for almost all x G fto-

Remark 1.5. In particular, if Si = T^, the commutant of the representation con-

sists of all decomposable operators

re

(1.34) T = / T(x) d/x(x)

such that

(1.35) T Or) U(x) = U (x) T (a (x))

for almost all x G ft, and the center of the representation consists of all decompos-

able operators

(1.36) T = / X(x)tn(x)dfi(x)

where the scalar function A G L°° (ft,d/i) is cr-invariant. Thus the representation

is a factor representation if and only if the right shift on L°° (ft,d/i) is ergodic. If

in addition dim (H (x)) — 1 for almost all x, then the representation is irreducible

since (1.35) then only has the trivial solutions T (x) = const.

Note that if the right shift on L2 (ft, dji) is ergodic, then dim (H (x)) is constant

for almost all x, and if Ho is a Hilbert space of that dimension, then we have an

isomorphism

r®

/ H {x) d/x (x) ^

L2

(ft, dfi) ® H0

and U may be viewed as a measurable function from ft into the unitary group

U (Ho) on Ho- The element T is then a function from ft into B (Ho) and (1.35)

takes the form

T (x) = U (x)T (a (x))U* (x).

Thus the commutant of the representation is canonically isomorphic to the fixed-

point algebra in

L°° (ft, dp) 8 B (Ho) = L°° (ft, B (Ho))