x m
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 dynamics—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
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