Symbolic Dynamical Systems and C*-Algebras: Continuous Orbit Equivalence of Topological Markov Shifts and Cuntz–Krieger Algebras

Preface
Acknowledgements
Contents
1 Introduction
2 Topological Markov Shifts
	2.1 Subshifts and Sliding Block Codes
		2.1.1 Full Shifts and Subshifts
		2.1.2 Higher Block Shifts
		2.1.3 Sliding Block Code
	2.2 Topological Markov Shifts
		2.2.1 Vertex Shifts
		2.2.2 Edge Shifts
		2.2.3 Shifts of Finite Type
	2.3 State Splitting and State Amalgamation
	2.4 Williams's Theorem
		2.4.1 Classification of Two-Sided Topological Markov Shifts
		2.4.2 Classification of One-Sided Topological Markov Shifts
	2.5 Appendix
		2.5.1 Shift Equivalence
		2.5.2 Perron–Frobenius Theorem
		2.5.3 Dimension Group
		2.5.4 Bowen–Franks Group and Smith Normal Form
		2.5.5 Topological Entropy and Zeta Function
	2.6 Notes
	References
3 Flow Equivalence
	3.1 Continuous Flow and Parry–Sullivan Theorem
	3.2 Bowen–Franks Group and Parry–Sullivan Determinant
	3.3 Franks's Theorem
	3.4 Ordered Cohomology Group
	3.5 Boyle–Handelman Theorem
	3.6 Notes
	References
4 Continuous Orbit Equivalence
	4.1 Continuous Orbit Equivalence
	4.2 Eventually Periodic Points
	4.3 Continuous Full Group
	4.4 Continuous Orbit Equivalence and Continuous Full Group
	4.5 Spatial Realization Theorem
	4.6 Inverse Semigroup
	4.7 Notes
	References
5 Étale Groupoids and Cuntz–Krieger Algebras
	5.1 Étale Groupoids
	5.2 Groupoid C*-Algebras
		5.2.1 Amenability
		5.2.2 Essential Principalness
		5.2.3 Minimality
		5.2.4 Pure Infiniteness
	5.3 The Étale Groupoid GA
	5.4 The C*-Algebra C*(GA)
	5.5 The Cuntz–Krieger Algebra mathcalOA
	5.6 The Maximal Commutative C*-Subalgebra mathcalDA
	5.7 Notes
	References
6 K-Theory for Infinite Simple C*-Algebras
	6.1 K-Theory for C*-Algebras
		6.1.1 K0-Group
		6.1.2 K1-Group
	6.2 Infinite Projections in Unital Simple C*-Algebras
	6.3 Purely Infinite C*-Algebras
	6.4 Ext-Groups for C*-Algebras
		6.4.1 Extensions of C*-Algebras
		6.4.2 Busby Invariant
		6.4.3 Pullback
		6.4.4 Equivalences
		6.4.5 Additive Structure
		6.4.6 Inverse
		6.4.7 The Ext-Groups Ext*(mathcalA)
	6.5 Notes
	References
7 K-Theory for Cuntz–Krieger Algebras
	7.1 Pure Infiniteness of Cuntz–Krieger Algebras
	7.2 K-Theory for Cuntz–Krieger Algebras
		7.2.1 K-Group for AF-Algebra mathcalFA
		7.2.2 K-Groups K*(mathcalOA) for Cuntz–Krieger Algebra mathcalOA
		7.2.3 The Group mathbbZN/ (1 - At)mathbbZN
		7.2.4 Examples
	7.3 Ext-Groups for Cuntz–Krieger Algebras
		7.3.1 Brief Review of Extension Groups
		7.3.2 Ext-Groups for Cuntz–Krieger Algebras
		7.3.3 The Homomorphism ιA: mathbbZrightarrowExts(mathcalOA)
		7.3.4 Examples
	7.4 Notes
	References
8 Strong Shift Equivalence, Flow Equivalence and Cuntz–Krieger Algebras
	8.1 Morita Equivalence of C*-Algebras
		8.1.1 Multiplier Algebras
		8.1.2 Imprimitivity Bimodules and Morita Equivalence
	8.2 Relative Morita Equivalence
		8.2.1 Relative σ-Unital C*-Algebras
		8.2.2 Relative Imprimitivity Bimodules and Relative Morita Equivalence
		8.2.3 Isomorphism of Relative Stabilizations
		8.2.4 Relative Full Corners
	8.3 Strong Shift Equivalence, Flow Equivalence and Cuntz–Krieger Algebras
		8.3.1 Corner Isomorphic Cuntz–Krieger Pairs
		8.3.2 Strong Shift Equivalence
		8.3.3 Isomorphism Φ*: K0(mathcalOA)rightarrowK0(mathcalOB)
		8.3.4 Flow Equivalence and Cuntz–Krieger Algebras
	8.4 Notes
	References
9 Classification Theorem for Continuous Orbit Equivalence
	9.1 Ordered Cohomology and Groupoid Cohomology
		9.1.1 One-Sided Ordered Cohomology Group
		9.1.2 The Groupoid Cohomology H1(GA)
	9.2 Continuous Orbit Equivalence and Groupoid Isomorphism
	9.3 Continuous Orbit Equivalence and Ordered Cohomology
	9.4 Finitely Presented Isomorphisms
	9.5 K-Group K0(mathcalOA) and Flow Equivalence
	9.6 Proof of the Classification Theorem
		9.6.1 (5) -3mu (6)
		9.6.2 (6) -3mu (7)
		9.6.3 (6) -3mu (8)
	9.7 Notes
	References
10 Gauge Actions and Continuous Orbit Equivalence
	10.1 Generalized Gauge Actions and Continuous Orbit Equivalence
	10.2 Strongly Continuous Orbit Equivalence
	10.3 One-Sided Eventual Conjugacy
	10.4 One-Sided Topological Conjugacy
	10.5 Examples
	10.6 Subequivalence Relations in Continuous Orbit Equivalence
	10.7 Cocycle Full Groups and Relative Continuous Orbit Equivalence
	10.8 Notes
	References
11 Classification Theorem for Flow Equivalence and Topological Conjugacy
	11.1 Classification Theorem for Flow Equivalence
	11.2 Kakutani Equivalence for Groupoids
	11.3 Proof of the Classification Theorem of Flow Equivalence
		11.3.1 Equivalence (1) -3mu (8) in Theorem 11.1.1
		11.3.2 Proof of Theorem 11.1.1
	11.4 Topological Conjugacy of Two-Sided Topological Markov Shifts
		11.4.1 Stabilization of One-Sided Topological Markov Shifts
		11.4.2 Two-Sided Conjugacy
	11.5 Transpose Free Isomorphisms of Cuntz–Krieger Triplets
		11.5.1 Out-Splitting
		11.5.2 In-Splitting
		11.5.3 Transpose Free Isomorphic Cuntz–Krieger Triplets
	11.6 Notes
	References
Index