INTRODUCTION

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a certain class of representations of the Cuntz algebras, and the chapter is moti-

vated partly by the representations arising from wavelet theory. These chapters

describe basic theory of Cuntz-relation representations which has also been consid-

ered in [71, Section 2.5.3]. Chapter 3 is devoted to so-called KMS states (short

for Kubo-Martin-Schwinger states) from statistical physics. More specifically, we

look at KMS states of the dynamic defined by one-parameter groups of quasi-free

automorphisms on a Cuntz C*-algebra. One theorem gives a characterization of

when there is, or is not, such a KMS state. Chapter 4 determines when, in this

situation, the fixed-point algebra is AF, and gives an outline of how to go about

computing the corresponding Bratteli diagrams. Since a main aim of the memoir is

to make an otherwise technical subject concrete, many examples are offered to help

the reader appreciate the abstract theorems, and to motivate the numerical invari-

ants that are introduced later. The rest of the memoir is devoted to the analysis of

the isomorphism classes of these AF-algebras. Thus the initial chapters are mainly

devoted to setting up the class of AF-algebras to be considered. But we feel that

they are of independent interest, and have relevance outside the particular issue of

AF-algebra classification. Part A is representation-theoretic, while Part B describes

numerical invariants for the AF-algebra classification context, and associated Brat-

teli diagrams. There are discussions of special cases of the general problem which

are related to: (i) the Williams problem (the issue of understanding the two main

ways of classifying dimension groups in dynamicsâ€”see [92] for more on this con-

nection); (ii) Krieger's theorem and conjugacy of the corresponding actions on the

Cuntz algebras (see [63]); (iii) invariants related to the Perron-Frobenius eigen-

value; and (iv) more algebraic invariants with examples, or situations where the

invariants become complete.

The one-parameter groups of automorphisms that we consider act upon the

Cuntz algebras, and they are called gauge actions; they scale the generators with a

unitary gauge. Our analysis is focused on the corresponding fixed-point algebras:

their classification, and their significance in, for example, symbolic dynamics. But

there are analogous and interesting questions for more general C*-algebras which

arise in dynamical systems, for instance the Cuntz-Krieger algebras OA- These

are algebras, naturally generalizing the Cuntz algebras, defined on generators Si

and relations, the relations being given by a 0-1 matrix A. The matrix A in turn

determines, in a standard fashion, see [63], a state space SA for a dynamical system

called a subshift. While we will not treat them here, our methods suggest directions

for future work on other gauge actions, as illustrated by recent papers of Exel, Laca,

and Vershik, for example [43] and [41, 42]. A gauge action on an algebra scales the

generators by a unitary phase, and a recent paper [40] defines the notions of gauge

action and KMS states in the setting of Cuntz-Krieger algebras. While our present

analysis for the Cuntz algebras is relatively managable for computations, involving

only Perron-Frobenius theory for finite matrices, the analogous analysis of Exel if

carried through would instead be based on an infinite-dimensional version of the

Perron-Frobenius theorem, due to Ruelle (see [82] and [83]). In Ruelle's theorem,

the finite matrix from the classical Perron-Frobenius theorem is replaced by an

operator, now called the Ruelle transfer operator (see [54], [12, Section 3.1]), and

the Perron-Frobenius left eigenvector from the classical case, by a measure on the

state space SA- The Ruelle transfer operator acts on functions on the compact

space SAI and the right Perron-Frobenius eigenvector is a continuous function on