The origin of this monograph was the desire to understand certain concrete
operator relations arising as realizations of filtering processes in signal theory. Our
research did, however, lead us naturally in the direction of analyzing certain non-
commutative dynamical systems and their fixed points and states. In this introduc-
tion, we give an overview of the contents of the monograph in three stages: First
(1) a very general discussion, then (2) a discussion with more specific definitions
and details, and finally (3) a detailed technical account of what the paper actually
1. General discussion and motivation. This monograph is centered around
the issue of distinguishing a particular family of AF-algebras, those which arise as
the centralizers of certain states on Cuntz algebras—or equivalently, as the fixed-
point algebras under certain one-parameter subgroups of the gauge action. (The
Cuntz algebras are the range of a functor from Hilbert space into C*-algebras. The
term AF-algebra is short for approximately finite-dimensional C*-algebra. Both
the Cuntz algebras and the AF-algebras play a role in several areas of mathemat-
ics, e.g., K-theory and dynamical systems, and in applications, for example to
statistical mechanics [19].) While AF-algebras generally are classified up to stable
isomorphism by the equivalence classes of their Bratteli diagrams, or by the iso-
morphism classes of their ordered dimension groups, none of the three items in this
triple of equivalence classes, of AF-algebras, Bratteli diagrams, or ordered dimen-
sion groups, respectively, is especially amenable to computation. In this memoir,
we therefore try to approach the subject via classes of concrete examples, which
do in fact admit algorithms for distinguishing isomorphism classes, in particular
by the computation of numerical invariants which distinguish these classes. The
examples are chosen so they illustrate in a concrete manner the main issues of
computation in each of the three incarnations, AF-algebras, Bratteli diagrams, or
dimension groups. The terms in this paragraph will be defined in Section 2 of this
The first part of our memoir is in a sense divorced from the central theme, that
of numerical invariants. But we feel that it helps the reader see how the main issue
fits into a wider context, and hopefully it will help the reader make connections to
the areas of mathematics and its applications which are touched upon in a more
indirect manner. The ubiquitous Cuntz algebras and their representations have a
surprising number of applications in a wide range of areas, also outside the subject
of operator algebras, such as wavelets and dynamical systems, to mention just two.
One of the recent areas of application of dimension groups and Perron-Frobenius
structures is to notions of equivalence for symbolic dynamics, and more specifically
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