Titlebook: Crossed Products of C*-Algebras, Topological Dynamics, and Classification; Thierry Giordano,David Kerr,Andrew Toms,Francesc P Textbook 201

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Internal Measure-Theoretic Phenomenae. It plays a pivotal role not only in combinatorial and geometric group theory but also in the theory of operator algebras through its various linear manifestations like hyperfiniteness, semidiscreteness, injectivity, and nuclearity. In this section we will review the theory of amenability for disc

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Internal Topological Phenomenaw, for discrete groups the topological notion of perturbation is trivial and thus, unlike the combinatorial measure-theoretic viewpoint, does not give us anything new beyond the merely group-theoretic. The group . is said to be . if every finite subset of . generates a finite subgroup. Equivalently,

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Phoebe Koundouri,Nikos A. PapandreouHere, we introduce the basic theory of the Cuntz semigroup. Any proofs not contained in these notes can be found in Ara–Brown–Guido–Lledo–Perera–Toms [1], but no originality is claimed there. Most of the results here are due to Cuntz [9], Kirchberg–R?rdam [17], and R?rdam [26]. We assume that all .*-algebras are separable unless otherwise noted.

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Graduate Students Becoming ScientistsThis is the first of three sections devoted to examples of group actions..In this section, we give examples of actions on commutative C*-algebras. In Section 8.2 we give a variety of examples of actions on noncommutative C*- algebras, and in Section 8.3 we give an additional collection of examples of actions that are similar to gauge actions.

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Nuts and Bolts: Proof SketchesIn this section we take some of the salient results of the preceding section and show more or less how they are proved. We shall take for granted, however, Winter’s locally finite nuclear dimension theorem; a full proof is well beyond the scope of these notes.

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External Measure-Theoretic PhenomenaAs discussed in Section 14.1, for discrete groups the basic idea of internal measuretheoretic finite approximation is captured by the F?lner set characterization of amenability. At the same time we can view F?lner sets as furnishing external finite approximations in the following way. Let . be a nonempty finite subset of a discrete group ..