Operator Algebras And Dynamics Groupoids Crossed Products And Rokhlin Dimension
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Operator Algebras and Dynamics: Groupoids, Crossed Products, and Rokhlin Dimension
This book collects the notes of the lectures given at the Advanced Course on Crossed Products, Groupoids, and Rokhlin dimension, that took place at the Centre de Recerca Matemàtica (CRM) from March 13 to March 17, 2017. The notes consist of three series of lectures. The first one was given by Dana Williams (Dartmouth College), and served as an introduction to crossed products of C*-algebras and the study of their structure. The second series of lectures was delivered by Aidan Sims (Wollongong), who gave an overview of the theory of topological groupoids (as a model for groups and group actions) and groupoid C*-algebras, with particular emphasis on the case of étale groupoids. Finally, the last series was delivered by Gábor Szabó (Copenhagen), and consisted of an introduction to Rokhlin type properties (mostly centered around the work of Hirshberg, Winter and Zacharias) with hints to the more advanced theory related to groupoids.
Ergodic Theory
This volume in the Encyclopedia of Complexity and Systems Science, Second Edition, covers recent developments in classical areas of ergodic theory, including the asymptotic properties of measurable dynamical systems, spectral theory, entropy, ergodic theorems, joinings, isomorphism theory, recurrence, nonsingular systems. It enlightens connections of ergodic theory with symbolic dynamics, topological dynamics, smooth dynamics, combinatorics, number theory, pressure and equilibrium states, fractal geometry, chaos. In addition, the new edition includes dynamical systems of probabilistic origin, ergodic aspects of Sarnak's conjecture, translation flows on translation surfaces, complexity and classification of measurable systems, operator approach to asymptotic properties, interplay with operator algebras
Symbolic Dynamical Systems and C*-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.