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),
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