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E-grāmata: Crossed Products of C*-Algebras, Topological Dynamics, and Classification

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Crossed Products of C*-Algebras, Topological Dynamics, and ClassificationPalielināt
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This book collects the notes of the lectures given at an Advanced Course on Dynamical Systems at the Centre de Recerca Matemàtica (CRM) in Barcelona. The notes consist of four series of lectures.

The first one, given by Andrew Toms, presents the basic properties of the Cuntz semigroup and its role in the classification program of simple, nuclear, separable C*-algebras. The second series of lectures, delivered by N. Christopher Phillips, serves as an introduction to group actions on C*-algebras and their crossed products, with emphasis on the simple case and when the crossed products are classifiable. The third one, given by David Kerr, treats various developments related to measure-theoretic and topological aspects of crossed products, focusing on internal and external approximation concepts, both for groups and C*-algebras. Finally, the last series of lectures, delivered by Thierry Giordano, is devoted to the theory of topological orbit equivalence, with particular attention to the classification of minimal actions by finitely generated abelian groups on the Cantor set.

I The Cuntz Semigroup and the Classification of C*-Algebras
1(38)
Andrew S. Toms
1 Introduction
3(2)
2 The Cuntz Semigroup
5(6)
2.1 The Category Cu
8(1)
2.2 Functional and Cu
9(2)
3 Structure of the Cuntz Semigroup
11(6)
4 Elliott's Program
17(4)
4.1 Nuclearity
17(2)
4.2 Counterexamples
19(2)
5 A New Conjecture
21(8)
5.1 Z-Stability
21(1)
5.2 Strict Comparison
22(2)
5.3 Nuclear Dimension
24(1)
5.4 The Conjecture: Principle and Progress
24(5)
6 Nuts and Bolts: Proof Sketches
29(10)
6.1 Proof Sketch for Theorem 5.4.7
29(2)
6.2 Proof Sketch for (a Special Case of) Theorem 5.4.8
31(4)
Bibliography
35(4)
II An Introduction to Crossed Product C*-Algebras and Minimal Dynamics
39(310)
N. Christopher Phillips
7 Introduction and Motivation
41(10)
8 Group Actions
51(42)
8.1 Examples of Group Actions on Locally Compact Spaces
51(14)
8.2 Examples of Group Actions on Noncommutative C*-Algebras
65(18)
8.3 Additional Examples of Generalized Gauge Actions
83(10)
9 Group C*-algebras and Crossed Products
93(76)
9.1 C*-Algebras of Discrete Groups
93(22)
9.2 Simplicity of the Reduced C*-Algebra of a Free Group
115(4)
9.3 C*-Algebras of Locally Compact Groups
119(10)
9.4 Crossed Products
129(10)
9.5 Reduced Crossed Products
139(12)
9.6 Computation of Some Examples of Crossed Products
151(18)
10 Some Structure Theory for Crossed Products by Finite Groups
169(52)
10.1 Introductory Remarks on the Structure of C*-Algebras
169(14)
10.2 Crossed Products by Finite Groups
183(5)
10.3 The Rokhlin Property for Actions of Finite Groups
188(16)
10.4 The Tracial Rokhlin Property for Actions of Finite Groups
204(17)
11 Crossed Products by Minimal Homeomorphisms
221(54)
11.1 Minimal Actions and Their Crossed Products
221(11)
11.2 Classifiability: Introduction and a Special Case
232(15)
11.3 Minimal Homeomorphisms of Finite-Dimensional Spaces
247(28)
12 Large Subalgebras and Applications to Crossed Products
275(74)
12.1 The Cuntz Semigroup
275(7)
12.2 Large Subalgebras
282(8)
12.3 Basic Properties of Large Subalgebras
290(8)
12.4 Large Subalgebras and the Radius of Comparison
298(10)
12.5 Large Subalgebras in Crossed Products by Z
308(7)
12.6 Application to the Radius of Comparison of Crossed Products
315(9)
12.7 Open Problems on Large Subalgebras
324(5)
Bibliography
329(20)
III C*-Algebras and Topological Dynamics: Finite Approximation and Paradoxicality
349(84)
David Kerr
13 Introduction
351(10)
14 Internal Measure-Theoretic Phenomena
361(20)
14.1 Amenable Groups and Nuclearity
361(4)
14.2 Amenable Actions, Nuclearity, and Exactness
365(3)
14.3 The Type Semigroup, Invariant Measures, and Pure Infiniteness
368(8)
14.4 The Universal Minimal System
376(3)
14.5 Minimal Actions, Pure Infiniteness, and Nuclearity
379(2)
15 External Measure-Theoretic Phenomena
381(12)
15.1 Sofic Groups, Sofic Actions, and Hyperlinearity
381(2)
15.2 Entropy
383(5)
15.3 Combinatorial Independence
388(2)
15.4 Mean Dimension
390(3)
16 Internal Topological Phenomena
393(20)
16.1 Locally Finite Groups and AF Algebras
393(1)
16.2 Dimension and K-Theoretic Classification
394(5)
16.3 Minimal Homeomorphisms of Zero-Dimensional Spaces
399(3)
16.4 Minimal Homeomorphisms of Finite-Dimensional Spaces
402(4)
16.5 Mean Dimension and Comparison in the Cuntz Semigroup
406(7)
17 External Topological Phenomena
413(20)
17.1 Groups Which Are Locally Embeddable into Finite Groups
413(3)
17.2 Chain Recurrence, Residually Finite Actions, and MF Algebras
416(9)
Bibliography
425(8)
IV Minimal Topological Systems and Orbit Equivalence
433(60)
Thierry Giordano
18 Introduction
435(2)
19 Cantor Dynamics
437(8)
19.1 Definitions of Some Dynamical Concepts
437(1)
19.2 Cantor Minimal Systems
438(1)
19.3 Dynamic and Ordered Bratteli Diagrams
438(7)
19.3.1 Bratteli Diagrams
438(2)
19.3.2 Ordered Bratteli Diagrams and the Bratteli--Vershik Model
440(5)
20 Etale Equivalence Relations
445(18)
20.1 Etale Equivalence Relations
445(3)
20.2 Isomorphism and Orbit Equivalence
448(1)
20.3 AF-Relations
449(3)
20.4 Invariants of Etale Equivalence Relations
452(9)
20.4.1 Pre-Ordered and Ordered Groups
452(5)
20.4.2 Dimension Groups
457(1)
20.4.3 Pre-Ordered Groups Associated to an Etale Equivalence Relation
458(3)
20.5 The Bratteli--Elliott--Krieger Theorem
461(2)
21 The Absorption Theorem
463(4)
22 Orbit Equivalence of AF-Equivalence Relations
467(4)
23 Orbit Equivalence of Minimal Actions of Abelian Groups
471(8)
23.1 Orbit Equivalence of Minimal Z-Actions on the Cantor Set
472(1)
23.2 Orbit Equivalence of Minimal Actions of Abelian Groups
473(2)
23.3 A Topological Krieger Theorem: Strong Orbit Equivalence
475(4)
24 Orbit Realization: Ormes' Results
479(6)
24.1 Orbit Realization for Minimal Homeomorphisms
479(1)
24.2 Orbit Equivalence of a Cantor Minimal System and its (Continuous) Spectrum
480(1)
24.3 Strong Orbit Equivalence for Minimal Homeomorphisms
481(4)
25 Full Groups
485(8)
25.1 The Measurable Case
485(3)
25.1.1 Algebraic Properties of [ ***R]
487(1)
25.1.2 The Full Group as a Topological Group
487(1)
25.2 The Topological Case
488(5)
25.2.1 Topological Dye's Reconstruction Theorems
489(2)
25.2.2 The Full Group of a Cantor Minimal System as a Topological Group
491(2)
Bibliography 493
Thierry Giordano is a Professor at the University of Ottawa, Canada.

David Kerr is a Professor at the Texas A&M University in College Station, TX, USA.

N. Christopher Phillips is a Professor at the University of Oregon in Eugene, OR, USA.

Andrew S. Toms is a Professor at the Purdue University in West Lafayette, IN, USA.
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