2006/07, 6th semester

Course Materials
Equivariant KK-theory and noncommutative index theory

In noncommutative geometry, a C*-algebra is regarded as an algebra of continuous functions on a certain 'noncommutative space'. There are two natural topological invariants of such spaces. On the one hand, K-theory of C*-algebras extends the Atiyah-Hirzebruch K-theory of topological spaces. On the other hand, K-homology arises from the study of analysis on the noncommutative space. These two theories, which are dual to each other, find a natural unification in the KK-theory of Kasparov, which arises from a deep understanding of the Atiyah-Singer index theory.

The first part of this course will provide the necessary background, the definition and the fundamental properties of KK-theory. In particular, we shall cover the basic properties of Fredholm modules, and their characters, which are given by natural cyclic cocycles first defined by Alain Connes. Hilbert modules provide natural extension of Hilbert spaces and are a key technical notion required for the definition of KK-theory. A very important property of KK-theory, which makes it very useful in applications, is the existence of a composition-type product, which we shall discuss in some detail.

The second part of the course will introduce an equivariant version of KK-theory which will be very useful in the study of properties of group actions on noncommutative spaces.

The third part of the course will acquaint the student with the Baum-Connes conjecture. This hypothesis, which has generated intense interest and some beautiful results over the past two decades, serves as a very natural view point on the interaction between topology-differential geometry and the K-theory of C*-algebras.

Prerequisites for this course include basic functional analysis (operators on Hilbert space), C*-algebras and elementary topology and differential geometry.

Course Summary:

  1. Fredholm operators and Fredholm modules. Characters of Fredholm modules and cyclic theory.
  2. K-homology and equivariant K-homology.
  3. Hilbert C*-modules and Morita invariance.
  4. Kasparov's KK-theory.
  5. Crossed-product C*-algebras and equivariant KK-theory.
  6. Review of Spin^c structures and Dirac operators
  7. Families of Dirac-type operators
  8. Atiyah's proof of Bott periodicity
  9. The Baum-Connes conjecture
  10. The Chern character

Part I
Jacek Brodzki
Notes by P. Witkowski
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Table of contents
  1. Introduction to KK-theory (Christian Voigt)
    1. Motivation and background
    2. Definition
    3. Properties
    4. Applications and further development
  2. C*-algebras
    1. Definitions
    2. Examples
    3. Gelfand transform
  3. K-theory
    1. Definitions
    2. Unitizations and multiplier algebras
    3. Stabilization
    4. Higher K-theory
    5. Excision and relative K-theory
    6. Products
    7. Bott periodicity
    8. Cuntz's proof of Bott periodicity
    9. The Mayer-Vietoris sequence
  4. Hilbert modules
    1. Definitions
    2. Kasparov stabilization theorem
    3. Morita equivalence
    4. Tensor products of Hilbert modules
  5. Fredholm modules and Kasparov's K-homology
    1. Fredholm modules
    2. Commutator conditions
    3. Quantised calculus of one variable
    4. Quantised differential calculus
    5. Closed graded trace
    6. Index pairing formula
    7. Kasparov's K-homology
  6. Boundary maps in K-homology
    1. Relative K-homology
    2. Semisplit extensions
    3. Schrodinger pairs
    4. The index pairing
    5. Product of Fredholm modules
  7. Equivariant K-homology of spaces
  8. KK-theory
    1. Kasparov's bifunctor
    2. Equivariant KK-theory
    3. Kasparov's product
Part II
Paul F. Baum
Notes by P. Witkowski
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Table of contents
  1. KK-theory
    1. C*-algebras
    2. K-thory
    3. Representations
    4. K-homology
    5. Equivariant K-homology
    6. Hilbert modules
    7. Reduced crossed product
    8. Topological K-theory of G
    9. KK-theory
    10. Equivariant K-theory
    11. K-theory of the reduced group C*-algebra
    12. KK_G^0(C, C)
Exam Exam questions

The exam was on 12th June 2007. It consisted of the written part (five exercises) and oral part. In the oral part each student had to answer two questions: easy one and difficult one (chosen from the two difficult questions).

Two students (on the graduate level) passed the exam.

Exam, written and oral

Designed by: Pawel Witkowski