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 xi
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