Combinatorial Set Theory of C*-Algebras

ISBN-13: 9783030270919 / Angielski / Twarda / 2019 / 517 str.

Ilijas Farah

Recenzje

"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)

Spis treści

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.

O autorze

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).

Blurb

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 interplay between logic and operator algebras (C*-algebras, in particular) is relatively young and the author is at the forefront of this interaction. The deep level of scholarship contained in these pages is evident and 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 the book a convenient, self-contained source for students.  A fair number of the exercises form an integral part of the text. They are chosen to widen and deepen the material from the corresponding chapters. Some other exercises serve as a warmup for the latter chapters.