Combinatorial Set Theory of C*-algebras

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).

“The target audience is graduate students with background in functional analysis and rudiments of mathematical logic, but the book can be read, with some additional effort, by a well-motivated student who has never encountered either of the two subjects. The book can be used as a blueprint to design a graduate-level course in the applications of set theory to operator algebras, and contains hints for future directions of research.” (Alessandro Vignati, Mathematical Reviews, February, 2021)

• 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 C*-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.- Index.- List of Symbols.