Programme for the course
K-theory for C*-algebras
For a description of the course, details of when and where we meet, and how to register, please consult the official web page of SIS.
The course will largely follow the book "An introduction to K-theory for C*-algebras" by Mikael Rørdam, Flemming Larsen, and Niels Jakob Laustsen. The book has a home page containing, among other things, a list of corrections. Do check it! (And please let me know if you spot any further errors.)
To pass the course, at least 8 homework exercises must be handed in. Not all the exercises are straightforward; if you need a hint, please feel free to ask, either in person at the lectures or at my office (HCØ 04.2.14), or via e-mail.
If you want to be assessed by a grade ("karakter") rather than just pass/fail, then you must inform me of this in writing when you hand in your first homework exercise. For those who want grades a deadline of three weeks applies to handing in the homework exercises (so that for instance Exercise 2.5 set on 1st September must be handed in no later than at the lectures on 22nd September).
All exercises, whether assessed by a grade or not, must be handed in by noon on 16th December, 2004, unless you have very good reasons (in which case you should speak to me asap). However, I strongly recommend that you do the exercises on a regular basis during the semester!
| Date | Content (subject to change) | Homework |
|---|---|---|
| 1st Sept | Overview and background. Introduction: the K0-group of a unital ring (RLL, Chapter 1, Section 3.1, and Exercise 3.10). | RLL, Exercise 2.5 |
| 8th Sept | The K0-group of a unital ring (continued). Alternative definitions of the K0-group of a unital C*-algebra: Murray-von Neumann equivalence and unitary equivalence of projections (RLL, parts of Sections 3.1-3.2 and 2.2-2.3). | RLL, Exercise 3.11 |
| 15th Sept | Unitary equivalence of projections (continued). Homotopy equivalence (RLL, Section 2.1 and parts of 2.2). | RLL, Exercise 2.9 |
| 22nd Sept | More about homotopy equivalence. Examples (RLL, Sections 3.2 and 3.3). | RLL, Exercise 3.1 |
| 29th Sept | K0(C(X)) for certain compact Hausdorff spaces X. The K0-group of a non-unital C*-algebra (RLL, Sections 3.3 and 4.1). | RLL, Exercise 3.8 |
| 6th Oct | Functoriality and the standard picture of K0 in the non-unital case (RLL, part of Section 4.1 and Section 4.2). | RLL, Exercise 4.2 |
| 13th Oct | No teaching - autumn holidays. | - |
| 20th Oct | Exactness properties of K0 (RLL, Section 4.3). | RLL, Exercise 4.6 (when answering (vii), you may use anything you like from Exercise 4.5 without proof) |
| 27th Oct | The K1-group of a C*-algebra (RLL, Section 8.1). | RLL, Exercise 8.9 (part (iv) requires Exercise 4.6; the map omega is defined on p. 144) |
| 3rd Nov | K1 and invertible elements; functoriality of K1 (RLL, Prop. 2.1.8, Remark 8.1.7, and Section 8.2). | RLL, Exercises 8.2 and 8.4 (these two small exercises count as one!) |
| 10th Nov | The index map (RLL, Sections 9.1-9.2). | RLL, Exercise 8.16 |
| 17th Nov | The index map (continued) (RLL, Sections 9.2-9.3). | RLL, Exercise 9.3 |
| 24th Nov | The K-groups of the compact operators and other operator ideals, and the relation to Fredholm theory (notes (to be handed out) and/or RLL, Section 9.4). | Exercise C.4 in the notes |
| 1st Dec | The higher K-groups (RLL, Chapter 10). | RLL, Exercise 10.5 |
| 8th Dec | The Bott map and Bott periodicity (RLL, Sections 11.1-11.2). | RLL, Exercise 11.1 |
| 15th Dec | Applications of Bott periodicity, the exponential map, and the six term exact sequence (RLL, Section 11.3 and Chapter 12). | - |
| 16th Dec (12:00) | Deadline for handing in homework exercises. | - |