Ultrafilters Throughout Mathematics (Graduate Studies in Mathematics)

"Ultrafilters and ultraproducts provide a useful generalization of the ordinary limit processes which have applications to many areas of mathematics. Typically, this topic is presented to students in specialized courses such as logic, functional analysis, or geometric group theory. In this book, the basic facts about ultrafilters and ultraproducts are presented to readers with no prior knowledge of the subject and then these techniques are applied to a wide variety of topics. The first part of the book deals solely with ultrafilters and presents applications to voting theory, combinatorics, and topology, while also dealing also with foundational issues. The second part presents the classical ultraproduct construction and provides applications to algebra, number theory, and nonstandard analysis. The third part discusses a metric generalization of the ultraproduct construction and gives example applications to geometric group theory and functional analysis. The final section returns to more advanced topics of a more foundational nature. The book should be of interest to undergraduates, graduate students, and researchers from all areas of mathematics interested in learning how ultrafilters and ultraproducts can be applied to their specialty." Sommario tratto dal retro della copertina Cover Title page Preface Part 1. Ultrafilters and their applications Chapter 1. Ultrafilter basics 1.1. Basic definitions 1.2. The ultrafilter quantifier 1.3. The category of ultrafilters 1.4. The number of ultrafilters 1.5. The ultrafilter number u 1.6. The Rudin-Keisler order 1.7. Notes and references Chapter 2. Arrow’s theorem on fair voting 2.1. Statement of the theorem 2.2. The connection with ultrafilters 2.3. Block voting 2.4. Finishing the proof 2.5. Notes and references Chapter 3. Ultrafilters in topology 3.1. Ultralimits 3.2. The Stone-Čech compactification: the discrete case 3.3. z-ultrafilters and the Stone-Čech compactifications in general 3.4. The Stone representation theorem 3.5. Notes and References Chapter 4. Ramsey theory and combinatorial number theory 4.1. Ramsey’s theorem 4.2. Idempotent ultrafilters and Hindman’s theorem 4.3. Banach density, means, and measures 4.4. Furstenberg’s correspondence principle 4.5. Jin’s sumset theorem 4.6. Notes and references Chapter 5. Foundational concerns 5.1. The ultrafilter theorem and the axiom of choice: Part I 5.2. Can there exist a “definable” ultrafilter on N? 5.3. The ultrafilter game 5.4. Selective ultrafilters and P-points 5.5. Notes and references Part 2. Classical ultraproducts Chapter 6. Classical ultraproducts 6.1. Motivating the definition of ultraproducts 6.2. Ultraproducts of sets 6.3. Ultraproducts of structures 6.4. Łoś’s theorem 6.5. The ultrafilter theorem and the axiom of choice: Part II 6.6. Countably incomplete ultrafilters 6.7. Revisiting the Rudin-Keisler order 6.8. Cardinalities of ultraproducts 6.9. Iterated ultrapowers 6.10. A category-theoretic perspective on ultraproducts 6.11. The Feferman-Vaught theorem 6.12. Notes and references Chapter 7. Applications to geometry, commutative algebra, and number theory 7.1. Ax’s theorem on polynomial functions 7.2. Bounds in the theory of polynomial rings 7.3. The Ax-Kochen theorem and Artin’s conjecture 7.4. Notes and references Chapter 8. Ultraproducts and saturation 8.1. Saturation 8.2. First saturation properties of ultraproducts 8.3. Regular ultrafilters 8.4. Good ultrafilters: Part 1 8.5. Good ultrafilters: Part 2 8.6. Keisler’s order 8.7. Notes and references Chapter 9. Nonstandard analysis 9.1. Naïve axioms for nonstandard analysis 9.2. Nonstandard numbers big and small 9.3. Some nonstandard calculus 9.4. Ultrapowers as a model of nonstandard analysis 9.5. Complete extensions and limit ultrapowers 9.6. Many-sorted structures and internal sets 9.7. Nonstandard generators of ultrafilters 9.8. Hausdorff ultrafilters 9.9. Notes and references Chapter 10. Limit groups 10.1. Introducing the class of limit groups 10.2. First examples and properties of limit groups 10.3. Connection with fully residual freeness 10.4. Explaining the terminology: the space of marked groups 10.5. Notes and references Part 3. Metric ultraproducts and their applications Chapter 11. Metric ultraproducts 11.1. Definition of the metric ultraproduct 11.2. Metric ultraproducts and nonstandard hulls of metric spaces 11.3. Completeness properties of the metric ultraproduct 11.4. Continuous logic 11.5. Reduced products of metric structures 11.6. Notes and references Chapter 12. Asymptotic cones and Gromov’s theorem 12.1. Some group-theoretic preliminaries 12.2. Growth rates of groups 12.3. Gromov’s theorem on polynomial growth 12.4. Definition of asymptotic cones 12.5. General properties of asymptotic cones 12.6. Growth functions and properness of the asymptotic cones 12.7. Properness of asymptotic cones revisited 12.8. Nonhomeomorphic asymptotic cones 12.9. Notes and references Chapter 13. Sofic groups 13.1. Ultraproducts of bi-invariant metric groups 13.2. Definition of sofic groups 13.3. Examples of sofic groups 13.4. An application of sofic groups 13.5. Notes and references Chapter 14. Functional analysis 14.1. Banach space ultraproducts 14.2. Applications to local geometry of Banach spaces 14.3. Commutative C*-algebras and ultracoproducts of compact spaces 14.4. The tracial ultraproduct construction 14.5. The Connes embedding problem 14.6. Notes and references Part 4. Advanced topics Chapter 15. Does an ultrapower depend on the ultrafilter? 15.1. Statement of results 15.2. The case when M is unstable 15.3. The case when M is stable 15.4. Notes and references Chapter 16. The Keisler-Shelah theorem 16.1. The Keisler-Shelah theorem 16.2. Application: Elementary classes 16.3. Application: Robinson’s joint consistency theorem 16.4. Application: Elementary equivalence of matrix rings 16.5. Notes and references Chapter 17. Large cardinals 17.1. Worldly cardinals 17.2. Inaccessible cardinals 17.3. Measurable cardinals 17.4. Strongly and weakly compact cardinals 17.5. Ramsey cardinals 17.6. Measurable cardinals as critical points of elementary embeddings 17.7. An application of large cardinals 17.8. Notes and references Part 5. Appendices Appendix A. Logic A.1. Languages and structures A.2. Syntax and semantics A.3. Embeddings A.4. References Appendix B. Set theory B.1. The axioms of ZFC B.2. Ordinals B.3. Cardinals B.4. V and L B.5. Relative consistency statements B.6. Relativization and absoluteness B.7. References Appendix C. Category theory C.1. Categories C.2. Functors, natural transformations, and equivalences of categories C.3. Limits C.4. References Appendix D. Hints and solutions to selected exercises Bibliography Index Back Cover This book provides the main results and ideas in the theories of completely bounded maps, operator spaces, and operator algebras, along with some of their main applications. It requires only a basic background in functional analysis to read through the book. The descriptions and discussions of the topics are self-explained. It is appropriate for graduate students new to the subject and the field. The book starts with the basic representation theorems for abstract operator spaces and their mappings, followed by a discussion of tensor products and the analogue of Grothendieck's approximation property. Next, the operator space analogues of the nuclear, integral, and absolutely summing mappings are discussed. In what is perhaps the deepest part of the book, the authors present the remarkable “non-classical” phenomena that occur when one considers local reflexivity and exactness for operator spaces. This is an area of great beauty and depth, and it represents one of the triumphs of the subject. In the final part of the book, the authors consider applications to non-commutative harmonic analysis and non-self-adjoint operator algebra theory. Operator space theory provides a synthesis of Banach space theory with the non-commuting variables of operator algebra theory, and it has led to exciting new approaches in both disciplines. This book is an indispensable introduction to the theory of operator spaces. "Ultrafilters and ultraproducts provide a useful generalization of the ordinary limit processes which have applications to many areas of mathematics. Typically, this topic is presented to students in specialized courses such as logic, functional analysis, or geometric group theory. In this book, the basic facts about ultrafilters and ultraproducts are presented to readers with no prior knowledge of the subject and then these techniques are applied to a wide variety of topics. The first part of the book deals solely with ultrafilters and presents applications to voting theory, combinatorics, and topology, while also dealing also with foundational issues. The second part presents the classical ultraproduct construction and provides applications to algebra, number theory, and nonstandard analysis. The third part discusses a metric generalization of the ultraproduct construction and gives example applications to geometric group theory and functional analysis. The final section returns to more advanced topics of a more foundational nature. The book should be of interest to undergraduates, graduate students, and researchers from all areas of mathematics interested in learning how ultrafilters and ultraproducts can be applied to their specialty."-- Back cover Provides the main results and ideas in the theories of completely bounded maps, operator spaces, and operator algebras, along with some of their main applications. The descriptions and discussions of the topics are self-explained. The book is appropriate for graduate students new to the subject and the field.