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Operator algebras

Kasparov's KK-theory

Extensions, analytic K-homology, graded cycles and Kasparov products, followed by Bott and Thom equivalences, index theory, equivariance and assembly maps.

Take first

Assumes C*-algebras, operator K-theory and Hilbert C*-modules. Exact prerequisite results are stated and cited in the lessons.

Lessons

  1. Extensions of C*-algebras and the Busby invariant
  2. Ext groups, absorption and Brown–Douglas–Fillmore theory
  3. Fredholm modules and analytic K-homology
  4. The index pairing between K-theory and K-homology
  5. Graded C*-algebras, Clifford algebras and graded Hilbert modules
  6. Kasparov modules and the groups KK(A, B)
  7. Pictures of KK: Fredholm operators, quasihomomorphisms and extensions
  8. Kasparov's technical theorem
  9. Connections and the existence of the Kasparov product
  10. Homotopy, associativity, the index pairing and KK-equivalence
  11. Unbounded bivariant K-theory and correspondences in noncommutative geometry
  12. Bott periodicity in KK: the Bott and Dirac elements
  13. Thom isomorphisms and K-orientations in KK
  14. Exact sequences in KK and the universal coefficient theorem
  15. Dirac classes and the cotangent Dolbeault element
  16. Wrong-way maps for K-oriented maps
  17. Equivariant KK-theory and the Green–Julg theorem
  18. Descent and the K-theory of crossed products
  19. Proper actions, universal proper spaces and equivariant K-homology
  20. The Baum–Connes assembly map and the conjecture
  21. Asymptotic morphisms and E-theory
  22. Deformations and the analytic index in E-theory
  23. The longitudinal index theorem and assembly for foliations

Authors and status

Written by GPT-6.1 Sol (OpenAI).

All 23 planned numbered lessons are available.

References

Each lesson ends with its own list of references. See sources, attribution and component terms.

Licence

The independent lesson text, proofs, examples, exercises and original diagrams written for this course. It is dedicated under CC0 1.0: no copyright is claimed, and to the extent any right exists anywhere, it is waived. The matrix-positivity companion uses the same CC0 dedication. Font notices: DejaVu, STIX, BaKoMa. Works cited in the lessons keep their own licences.

Licensing on this site · LICENSING.md

The Markdown files of this course are in the GitHub repository.

Reading paths

Operator relations and their index: Extensions, analytic K-homology, index pairing, then absorption and planar classification.

Composition of correspondences: graded modules, KK groups, Fredholm pictures, technical partitions, connections and products, and homotopy and associativity, unbounded cycles and spectral triples, Bott periodicity and exact sequences and the universal coefficient theorem.

Bundles and symmetry: Thom symbols and orientations, wrong-way maps and their composition, equivariant cycles and Green–Julg, and descent and Dirac–dual Dirac. These lessons identify their exact analytic prerequisites.

Matrix positivity: complete positivity and matrix tests, with proofs and solved exercises.

Noncompact foundations for equivariant induction proves Haar existence, homogeneous quotients, cutoffs and Banach integration. It is included in the PDF, LaTeX and source ZIP.

Cohomological foundations for the complex Thom comparison proves the ordinary Thom theorem, projective splitting, relative even character and Todd comparison used in Lesson 13. It is included in the PDF, LaTeX and source ZIP.

Editable mathematical sources

Read these 23 lessons and three companions as a PDF · LaTeX for these 23 lessons and three companions · Editable source ZIP

The package contains the lessons currently available above, their individual LaTeX sources, original figure sources and reproduction instructions.