xiv REPRESENTATION THEORY AND NUMERICAL AF-INVARIANTS
SA Following the model from the simpler case of the Cuntz algebra, and using the
Ruelle-Perron-Frobenius theorem, Exel in [40] computes the KMS states for the
Cuntz-Krieger algebras, and it would be interesting to analyze the issues of fixed-
point subalgebras and state centralizers in this context. Relevant for our present
analysis is the fact that unital endomorphisms of B(7i) are known to correspond
to representations of the Cuntz algebras Od, and the integer d is the Powers index.
The corresponding index in the setting of [40] is not necessarily integer-valued,
but some methods in our memoir point to generalizations and future directions of
research in this and other algebraic settings of symbolic dynamics.
The use of certain representations of Od in wavelet theory is outlined in [12,
Exercises 1-11, 1-12, and 2-25]. The starting observations there are the facts
(i) that the system of frequency-subband filters with d subbands is known to satisfy
the relations which define representations of Od, and (ii) that the multiresolution
approach to wavelets is based on subband filters.
Yet other uses of the representations of the Cuntz algebras Od to dynamics
include the work of J. Ball and V. Vinnikov [1, 2] on Lax-Phillips scattering theory
for multivariable linear systems. These two papers develop a functional model for
representations of Od in a setting which is more special than that of Theorem 1.2
below.
The original AF-algebra in [6] had Pascal's triangle as its diagram and arose
as the fixed-point algebra of the standard infinite-product-type action of the circle
on the infinite tensor product of a sequence of full 2 x 2 matrix algebras. Infinite-
tensor-product-type actions of more general compact groups have been considered
by Anthony Wassermann, Adrian Ocneanu and others, and the fixed-point alge-
bras are related to the AF-algebras occuring in Vaughan Joness subfactor theory,
which again has important applications in knot theory, low-dimensional manifolds,
conformal field theory, etc.; see [38, 49]. But this is far outside the scope of the
present monograph.
As noted in [65], the list of applications of Bratteli diagrams includes such areas
of applied mathematics as the fast Fourier transform, and more general Fourier
transforms on finite or infinite nonabelian groups. These transforms in turn are
widely used in computational mathematics, in error-correction codes, and even in
P. Shor's fast algorithm for integer factorization on a quantum computer [84, 85].
As reviewed in [65], the effectiveness of the Bratteli diagrams for development of
algorithms pays off especially well when a sequence of operations is involved, with
a succession of steps, perhaps with a scaling similarity, and with embeddings from
one step into the next. In the case of matrix algebras, the embeddings are in the
category of rings, while for groups, the embeddings are group homomorphisms. The
number of embeddings may be infinite, as it is in the applications in the present
memoir, or it may be finite, as in the case of the fast-Fourier-transform algorithms
of Cooley and Tukey, see [65].
The discrete Fourier transform for the cyclic group Z / n Z involves the unitary
matrix - t
(V27rj/Cy/n)
. and is of complexity
0(n2).
If n is a composite number,
such as n =
2fc,
then there are choices of Bratteli diagrams built on Z / n Z which
turn the discrete Fourier transform into a fast Fourier transform of complexity
O(nlogn). For each of the Bratteli diagrams formed from a sequence of subgroups
of Z / n Z , the corresponding fast Fourier transform involves a summation over the
paths of the Bratteli diagram.
Previous Page Next Page