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Combinatorial Set Theory of C*-algebras

  • Book
  • © 2019

Overview

Authors:
  1. Ilijas Farah

    1. Department of Mathematics and Statistics, York University, Toronto, Canada

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  • First book on the applications of logic and set theory to operator algebras
  • Presents dramatic progress on the resolution of several long-standing problems on C*-algebras
  • Provides a concise guide to the relevant theory of sets and C*-algebras

Part of the book series: Springer Monographs in Mathematics (SMM)

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Table of contents (17 chapters)

  1. Front Matter

    Pages i-xxx
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  2. C∗-algebras

    1. Front Matter

      Pages 1-1
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    2. C∗-algebras, Abstract, and Concrete

      • Ilijas Farah
      Pages 3-46

    3. Examples and Constructions of C∗-algebras

      • Ilijas Farah
      Pages 47-78

    4. Representations of C∗-algebras

      • Ilijas Farah
      Pages 79-120

    5. Tracial States and Representations of C∗-algebras

      • Ilijas Farah
      Pages 121-136

    6. Irreducible Representations of C∗-algebras

      • Ilijas Farah
      Pages 137-174

  3. Set Theory and Nonseparable C∗-algebras

    1. Front Matter

      Pages 175-175
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    2. Infinitary Combinatorics I

      • Ilijas Farah
      Pages 177-197

    3. Infinitary Combinatorics II: The Metric Case

      • Ilijas Farah
      Pages 199-216

    4. Additional Set-Theoretic Axioms

      • Ilijas Farah
      Pages 217-239

    5. Set Theory and Quotients

      • Ilijas Farah
      Pages 241-273

    6. Constructions of Nonseparable C∗-algebras, I: Graph CCR Algebras

      • Ilijas Farah
      Pages 275-299

    7. Constructions of Nonseparable C∗-algebras, II

      • Ilijas Farah
      Pages 301-310

  4. Part III

    1. Front Matter

      Pages 311-311
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    2. The Calkin Algebra

      • Ilijas Farah
      Pages 313-336

    3. Multiplier Algebras and Coronas

      • Ilijas Farah
      Pages 337-348

    4. Gaps and Incompactness

      • Ilijas Farah
      Pages 349-365

    5. Degree-1 Saturation

      • Ilijas Farah
      Pages 367-392

    6. Full Saturation

      • Ilijas Farah
      Pages 393-420
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Keywords

About this book

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.

Reviews

“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)

Authors and Affiliations

  • Department of Mathematics and Statistics, York University, Toronto, Canada

    Ilijas Farah

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).

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