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.