Crossed products and groupoid C*-algebras

Crossed products, induced representations and covariant localization; duality and the Thom isomorphism; groupoid algebras, diagonals, ideals, Morita equivalence and geometric index constructions.

Seventeen lessons develop the constructions with proofs, worked examples and 77 exercises with complete solutions. Begin with C*-dynamical systems and proceed through induction and duality to groupoid algebras and geometric index constructions.

The supporting readings give the precise prerequisite proofs used by the course. The additional KK and E-theory assertions in Lessons 4, 12 and 17 are explicitly conditional on the bivariant prerequisites specified there.

  1. C*-dynamical systems and full crossed products
  2. Reduced crossed products and Fell's absorption principle
  3. Expectations, traces and ideals in crossed products
  4. Amenability and the equality of full and reduced crossed products
  5. Crossed products by integers and real numbers: Fourier coordinates and mapping tori
  6. Pontryagin duality and the dual action
  7. Induced algebras and Green's imprimitivity theorem
  8. Takai duality
  9. Proper actions, free actions and the orbit space
  10. The Connes–Thom isomorphism I: the Wiener–Hopf extension
  11. The Connes–Thom isomorphism II: surjectivity, naturality and consequences
  12. Exterior equivalence and Connes's construction of the Thom map
  13. Locally compact groupoids and Haar systems
  14. Groupoid C*-algebras: full and reduced
  15. Equivalence of groupoids and Morita equivalence of their C*-algebras
  16. C*-algebras of foliations and their Morita equivalences
  17. The tangent groupoid and deformation to the normal cone

Download the reader and editable sources · Sources, authorship and component terms · Result and source records

Written by GPT-6.1 Sol (OpenAI). Self-checked by the writing AI. Original course text: CC0 1.0.