Ilijas Farah: Massive C*-Algebras

11th January, 2021

This is a 22-lecture course, with each lecture being around 90 minutes, given by Ilijas Farah.

The route to understanding separable C*-algebras frequently involves a detour via non-separable C*-algebras, such as the Calkin algebra, the asymptotic sequence algebras, ultrapowers, and ultraproducts. Some basic ideas from logic can be used to analyse these massive C*-algebras. Among other things, we will see that the existence of outer automorphisms of the Calkin algebra depends on the set-theoretic axioms.

Prerequisites: The first course in functional analysis, with some acquaintance with operator theory and C*-algebras. No set-theoretic assumptions will be imposed.

Textbook: The recommended textbook is Combinatorial Set Theory of C*-algebras, Ilijas Farah, Springer Monographs in Mathematics, 2019.

  1. The strict topology. The multiplier algebra. Introducing coronas, 1
  2. The strict topology. The multiplier algebra. Introducing coronas, 2
  3. The corona of every σ-unital C*-algebra is countably degree-1 saturated
  4. Applications and limitations of countable degree-1 saturation
  5. Ultrapowers, Łoś’s Theorem, countable saturation
  6. Saturation, tracial ultrapowers
  7. Preliminary analysis of C*-algebras of density character ℵ1, Downward Lowenheim-Skolem Theorem
  8. All ultrapowers of a fixed separable C*-algebra are isomorphic
  9. Continuum Hypothesis implies that the Calkin algebra has outer automorphisms, 1
  10. Continuum Hypothesis implies that the Calkin algebra has outer automorphisms, 2
  11. Automorphisms of coronas, general remarks on forcing axioms
  12. Introducing OCA
  13. OCA implies every subset of Partℕ of cardinality ℵ1 is bounded
  14. OCA implies that every coherent family of unitaries is trivial
  15. Some continuous functional calculus and meager sets in product spaces
  16. Discretization of the space 𝒟[E] and liftings
  17. The isometry trick
  18. Stabilizers done right and Ulam stability
  19. From C-measurable ε-approximations to innerness and OCA∞
  20. Obtaining σ-narrow liftings
  21. OCA implies all automorphisms of the Calkin algebra are inner
  22. More on ultrapowers and asymptotic sequence algebras

These videos were produced by the Fields Institute, as a graduate course (link to course page).