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This book explores and highlights the fertile interaction between logic and operator algebras, which in recent years has led to the resolution of several long-standing open problems on C*-algebras. The author is at the forefront of research on the interplay between logic and operator algebras (C*-algebras, in particular). The deep level of scholarship contained in these pages opens doors to operator algebraists interested in learning about the set-theoretic methods relevant to their field, as well as to set-theorists interested in expanding their view to the non-commutative realm of operator algebras. Enough background is included from both subjects to make this book a convenient, self-contained source for students. Exercises form an integral part of the text. They widen and deepen the material from the corresponding chapters. Other exercises serve as a warmup for the latter chapters.
This second edition furthers the aims of the first, improving proofs and providing new insights into the relationship between logic and operator algebras. It has been expanded, adding new lemmas, definitions, propositions, and exercises. The book provides a comprehensive analysis of ultraproducts of tracial von Neumann algebras. It also introduces Kirchberg s epsilon-test for ultraproducts and reduced products, along with its strict strengthening.
List of contents
Preface.- Part I C*-algebras.- 1. C*-algebras, Abstract and Concrete.- 2. Examples and Constructions of C*-algebras.- 3. Representations of C*-algebras.- 4. Tracial States and Representations of C*-algebras.- 5. Irreducible Representations of C*-algebras.- Part II Set Theory and Nonseparable C*-algebras.- 6. Infinitary Combinatorics, I.- 7. Infinitary Combinatorics, II: The Metric Case.- 8. Additional Set-Theoretic Axioms.- 9. Set Theory and Quotients.- 10. Constructions of Nonseparable C*-algebras, I: Graph CCR Algebras.- 11. Constructions of Nonseparable C*-algebras, II.- Part III Massive Quotient algebras.- 12. The Calkin Algebra.- 13. Multiplier Algebras and Coronas.- 14. Gaps and Incompactness.- 15. Degree-1 Saturation.- 16. Full Saturation.- 17. Automorphisms of Massive Quotient C*-Algebras.-Part IV Appendices.- A. Axiomatic Set Theory.- B. Descriptive Set Theory.- C. Functional Analysis.- D. Model Theory.- References.- List of Symbols.- Index.
About the author
Ilijas Farah is Professor of Mathematics at York University in Toronto, Canada. His research interests include set theory and logic, and their applications to operator algebras. Prof. Farah is a leading expert and renowned expositor in operator algebras that apply various concepts, tools, and ideas from logic and set theory to classification problems in operator algebras. In addition to being an invited speaker at the 2014 ICM, Prof. Farah is the recipient of the Sacks Prize (1997), the Governor General’s Gold Medal for one of the two best doctorates at the University of Toronto (1998), The Canadian Association for Graduate Studies/University Microfilms International Distinguished Dissertation Award (1998), the Dean’s award for outstanding research (2006), and Faculty of Science Excellence in Research Award (2017).
