2004/05, 1st semester

Course Materials
K-theory of operator algebras

In mathematics it is always important to classify objects in question, which is usually done by assigning to them some recognizable invariants. The K-functor gives a very powerful method of this kind that associates abelian groups to various mathematical objects, mostly algebras. In particular this approach is very fruitful in the context of algebras of linear transformations of Hilbert spaces or, more specifically, C*-algebras. The K-theory of operator algebras, developed in the last thirty years, is now a cornerstone of Connes. Noncommutative geometry, being deeply interrelated with other aspects of the latter theory (cyclic homology, index theorems, foliations). Via a theorem by Gelfand and Naimark, which identifies commutative C*-algebras with locally compact Hausdorff spaces, it also includes a version of K-theory for classical spaces, going far beyond the classical situation in many respects. Indeed, it allows for a meaningful treatment of certain singular spaces (e.g., spaces of leaves of a foliation) that have a trivial topology in the usual sense. It also leads to classification results for certain C*-algebras. Moreover, since C*-algebras are the natural language for quantum physics, K-theory also may give invariants of physical systems (e.g., in the quantum Hall effect). The objective of this lecture course is to present the most elementary aspects of the K-theory of C*-algebras, providing an entry to more advanced topics in noncommutative geometry to be considered in forthcoming courses.

Concretely, the contents of the lectures is as follows:

  1. Preparations about C*-algebras (assuming no previous knowledge, we will start from basic notions and arrive at results that are fundamental for K-theory. In particular, we consider projections and unitaries and equivalence relations between them).
  2. Definition of K0, proof of general properties (functoriality and stabilization properties), computation of K0 for elementary examples.
  3. Definition of K1, proof of general properties.
  4. Bott periodicity. After introducing the suspension functor S, we will see that K1 = K0 composed with S. Higher K-groups are defined by succesive composition with S. On the other hand, there is also a functorial isomorphism between K0 and K1 composed with S, leading to Bott periodicity.
  5. Six-term exact sequence. For any ideal J of a C*-algebra A, the exact sequence given by the factorization with respect to the ideal gives rise to a (circular) six-term exact sequence involving K0 and K1 of A, J and A=J. This sequence is the most effective tool for the computation of K-groups of C*-algebras. Other tools (Kuenneth formula, Pimsner-Voiculescu exact sequence) are mentioned. With these methods at hand, we will compute several examples (Toeplitz algebra, Cuntz algebras, certain examples of quantum spaces).
K-theory of operator algebras
Rainer Matthes,
Wojciech Szymański
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Table of contents
  1. Preliminaries on C*-algebras
    1. Basic definitions
    2. Spectral theory
    3. Matrix algebras and tensor products
    4. Examples and exercises
  2. Projections and unitaries
    1. Homotopy for unitaries
    2. Equivalence of projections
    3. Semigroups of projections
    4. Examples and exercises
  3. The K_0 group for unital C*-algebras
    1. The Grothendieck construction
    2. Definition of the K_0 group of a unital C*-algebra
    3. Examples and exercises
  4. K_0-group - the general case
    1. Definition of the K_0-functor
    2. Further properties
    3. Inductive limits. Continuity properties and stability of K_0
    4. Examples and exercises
  5. K_1-functor and the index map
    1. The K_1-functor
    2. Examples and exercises
  6. Bott periodicity and the exact sequence of K-theory
    1. Higher K-groups
    2. Bott periodicity
    3. The 6-term exact sequence
    4. Examples and exercises
  7. Tools for the computations of K-groups
    1. Crossed products, the Connes-Thom isomorphism and Pimsner-Voiculescu sequence
    2. The Mayer-Vietoris sequence
    3. The Kunneth formula
Graph C*-algebras
Rainer Matthes
Notes by P. Witkowski
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Table of contents
  1. Universal graph C*-algebras
  2. Computations of K-theory
  3. Proof of the main theorem
Exam Exam questions

The exam was on 20th January 2005. It consisted of the written part (six 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).

Five students (on the graduate and undergraduate level) passed the exam.

Written part

Oral part

Designed by: Pawel Witkowski