TT

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

contains.

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

introduction.

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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