xvi REPRESENTATION THEORY AND NUMERICAL AF-INVARIANTS
on the AF-algebra can be removed by putting more structure on the group and the
diagram.)
Recently there has been a fruitful interaction between the theory of dynamical
systems, analytic number theory, and C*-algebras. In [4], the authors show how
(3-KMS states may be used in understanding the Riemann zeta function, and vice
versa. In [89], [21], [37], [49], [39], [52], [55], and [60], certain dynamical systems
are used to generate new simple C*-algebras from the Cuntz algebras, and to better
understand the corresponding isomorphism classes of C*-algebras. The results in
Chapter 4 should be contrasted with results of Izumi [53] and Watatani [93] which
deal with crossed product constructions built from the Cuntz algebras Od- Here
we study the AF-subalgebras of Od formed from the one-parameter automorphism
groups of Chapter 3.
It follows from the definition of the CVrelations that they are well adapted to
d-multiresolutions of the kind used in wavelets and fractal analysis. The number d
represents the scaling factor of the wavelet. This viewpoint was exploited in recent
papers [16], [9], [10], and [28]. While the representations for these applications
are type I, the focus in the present memoir is type III representations of Od, and a
family of associated AF-C*-algebras 21^ (C Od for some d). These representations
arise from a modified version of the technique which we used in generating wavelet
representations. This starting point in fact yields a general decomposition result
for representations of Od which seems to be of independent interest. To understand
better the resulting decomposition structure, we will establish that the centraliz-
ers of these states are simple AF-algebras, and that the Bratteli diagrams have
stationary incidence matrices J of a special form given in (7.2). (In general the
centralizer of a state w o n a C*-algebra 21 is defined as the set of x G 21 such that
UJ (xy) = UJ (yx) for all y G 21. If u; is a KMS state with respect to a dynamical
one-parameter group r, as in (3.3), then this is equivalent to saying that x is a
fixed point for the dynamic a.) Clearly the rank of the corresponding dimension
group is an invariant, but it appeared at first sight that different matrices J and J'
would yield non-isomorphic AF-algebras 21j and 21j/ . This turns out not to be the
case, however, and the bulk of the memoir concerns numerical AF-invariants. It is
not easy to get invariants that discriminate the most natural cases of algebras 2lj
that arise from this seemingly easy family of AF-algebras. There is a connection
to subshifts in dynamical systems, but if subshifts are constructed, from J and
J7, say, we note that strong shift equivalence in the sense of Krieger is equivalent
to J = J', while isomorphism of the AF-algebras 21j and 21j turns out to be a
much more delicate problem; see also [13]. While we do not have a complete set
of numerical AF-invariants for our algebras 21j , we do find interesting subfamilies
of 21j' s which do in fact admit a concrete isomorphism labeling, and Part B of the
memoir concentrates on these cases. In contrast, we mention that [14] and [15] do
have general criteria for C*-isomorphism of the algebras 21j , but those conditions
are rather abstract in comparison with the explicit and numerical invariants which
are the focus of the present memoir. A complete algorithm for C*-isomorphism is
written out in [14, pp. 1651-1653 and corrigendum].
Let us now go into even more nitty-gritty technical details:
3. Detailed outline. The Cuntz algebra Od with generators Si and relations
s*Sj = Sijl, Yli=i
sisi
~ 1 contains a natural abelian subalgebra Td = C (Y\^° Z^),
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