1. GENERAL REPRESENTATIONS OF Od ON A SEPARABLE HILBERT SPACE 9
Furthermore
(1.42) (StZ\r,) = (S\SiT))
= f (^x)\(SzV)(x))dp(x)
Jn
= I
Xi
(x) p (x) (U (x)* £(x)\r, (a (x))) dy (x)
= f p(x)(U(xr^(x)\r](a(x)))dfi(x)
= I p{°i(y))(ufo(y)TStoto) \vto}Pto
to)"2
d»to
Jn
= f p(cn
(y)V1
(U to (y)T £ to (y)) \ v (y)) dp, (y),
Jn
and the expression (1.22) for S* follows.
If a = (aia2 •.. OLn) with a& G Zj, define
One verifies from (1.21) and (1.22) that
(1.44) Sa£ (x) =
Xai
(x)
X
*2 ((T (x))
Xan
K " 1 (*))
xp(x)p(v(x))...p(vn-1{x))
xU(x)U (r (x)) --U
(a71"1
(x)) £
{an
(x))
and
(1.45) S^{x) = p(aan {x))~l p{a(Xn_1G0in{x))~ --p(aai aan (x))'1
x U (cran (x))* U (cran_1aari (x))* ---U (aai - - aan (x))* £ (^ai ^
a i
(x)).
Combining (1.44)-(1.45) with the relations
a
Q n
a " ( x ) = a " - 1 ( x ) ,
(1.46) :
O-QI ^ a c r " ( x ) = X,
which are valid if x = (cci,..., a
n
, x
n
+i,...) , we obtain
(1.47) SaS*at;(x)=xa(x)Z(x),
where
( 1 - 4 8 ) Xoc (xi,X2,Xs,. . . ) = JaiXi£a
2
S2 * * ' ^ a
n
x
n
-
This proves firstly that
d
(1.49) ^ 5 ^ = 1;
2 = 1
and (1.41) and (1.49) show that Si t— 5^ is indeed a representation of the Cuntz
relations. Secondly, (1.47) shows that Vj maps onto the algebra of operators on 7i
of the form
(1.50) [ X(x)tn{x)dp(x)
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