Symbolic Dynamical Systems and C*-Algebras
Continuous Orbit Equivalence of Topological Markov Shifts and Cuntz–Krieger Algebras
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Beschreibung
This book presents the interplay between topological Markov shifts and Cuntz–Krieger algebras by providing notations, techniques, and ideas in detail. The main goal of this book is to provide a detailed proof of a classification theorem for continuous orbit equivalence of one-sided topological Markov shifts. The continuous orbit equivalence of one-sided topological Markov shifts is classified in terms of several different mathematical objects: the étale groupoids, the actions of the continuous full groups on the Markov shifts, the algebraic type of continuous full groups, the Cuntz–Krieger algebras, and the K-theory dates of the Cuntz–Krieger algebras. This classification result shows that topological Markov shifts have deep connections with not only operator algebras but also groupoid theory, infinite non-amenable groups, group actions, graph theory, linear algebras, K-theory, and so on. By using this classification result, the complete classification of flow equivalence in two-sided topological Markov shifts is described in terms of Cuntz–Krieger algebras. The author will also study the relationship between the topological conjugacy of topological Markov shifts and the gauge actions of Cuntz–Krieger algebras.
This book presents the interplay between topological Markov shifts and Cuntz–Krieger algebras by providing notations, techniques, and ideas in detail. The main goal of this book is to provide a detailed proof of a classification theorem for continuous orbit equivalence of one-sided topological Markov shifts. The continuous orbit equivalence of one-sided topological Markov shifts is classified in terms of several different mathematical objects: the étale groupoids, the actions of the continuous full groups on the Markov shifts, the algebraic type of continuous full groups, the Cuntz–Krieger algebras, and the K-theory dates of the Cuntz–Krieger algebras. This classification result shows that topological Markov shifts have deep connections with not only operator algebras but also groupoid theory, infinite non-amenable groups, group actions, graph theory, linear algebras, K-theory, and so on. By using this classification result, the complete classification of flow equivalence in two-sided topological Markov shifts is described in terms of Cuntz–Krieger algebras. The author will also study the relationship between the topological conjugacy of topological Markov shifts and the gauge actions of Cuntz–Krieger algebras.
Presents the interplay between topological Markov shifts and Cuntz–Krieger algebras Provides a detailed proof of a classification theorem Shows topological Markov shifts’ deep connections
Autor*in
Kengo Matsumoto
Themen in »Symbolic Dynamical Systems and C*-Algebras«
Symbolic Dynamical System C*-Algebra Topological Markov Shift Cuntz-Krieger Algebra Continuous Orbit Equivalence Flow Equivalence Etale Groupoid K-Theory
Stimmen zu »Symbolic Dynamical Systems and C*-Algebras«
“This is a carefully written and welcome contribution to the research literature on classification of shifts and its connection to operator algebras. It develops the sophisticated theory carefully and from a low starting point, and will be of great use to researchers in the field.” (Thomas B. Ward, zbMATH 1571.46001, 2026)
“The text serves to introduce both the graduate student and the researcher to these topics. ... All the key topics-symbolic dynamics, Cuntz-Krieger algebras … and inverse semigroups-are introduced in a manner that seems accessible to interested graduate students. In an effort to be self-contained, many results of independent interest are proven in full detail. ... Each chapter ends with historical notes ... .” (Kevin Aguyar Brix, Mathematical Reviews, January, 2026)
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