10

REPRESENTATION THEORY AND NUMERICAL AF-INVARIANTS

where A ranges over all continuous functions on the Cantor set ft. Thus the restric-

tion of the representation s^ »— • Si to Vd is indeed the spectral representation.

To show the main part of Theorem 1.2, i.e., the existence of the objects Ti (x),

d/j, (x), U (x), one does indeed start with a spectral measure \i for the restriction of

the representation to Vd. The spectrum of Vd is ft, so this gives the decomposition

(1.17), and the action of Vd on U is given by (1.47). If / G C (ft) = Vd, and Mf is

the representative of / in Ti:

(1.51) Mf= J f(x)ln{x)dfjL(x),

then

d

(1.52) Mfoa = J2^MfS:

i=l

and the quasi-invariance of /x under a follows. Thus one may define p (x) by (1.23).

Similarly, if / G C (ft) = Vd has support in Oi (ft) = ift = ft^, one verifies that

(1.53) Mfoa% = S*MfSt.

Thus the two representations of C(ft^) given by / \-^ Mf on MXH and / ^-

MfQ(Ti on 7 Y are unitarily equivalent. In particular, this means that dim (Ti (x)) =

dim (Ti((Ji (x))) for /i-almost all x, so the constancy of dim(Ti(x)) almost every-

where over the orbits of Ji,..., ad follows. But (1.4) then implies that dim (Ti (x))

is constant on a-orbits (actually the two forms of constancy are equivalent). Also

it follows from the unitary equivalence (1.53) that \i is quasi-invariant under Oi and

1 /9

that p{pi (x))"1 = ( d / t J^) } ) ) exists. See [68] or [71, Section 2.5.3] for details

on spectral multiplicity theory. Now, one may define a representation Sj H- Ti of

Od on H by

(1.54) (TiZ)(x)=Xi(x)p(x)Z((T{x)).

One checks that this is indeed a representation of Od by the first part of the proof,

and by the proof of (1.47) it follows that

(1.55) TaTa = SaSa

for all multi-indices a. Define an operator U by

d

(i.56) u = Y,SiT;.

2 = 1

Using the Cuntz relations in a standard manner, one checks that U is a unitary

operator, and

(1.57) Sz = XJT%

for i = 1,..., d. Putting

(1.58) ia := (iaict2 ... an),