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For the Focused Semester on KK-theory and its applications I'm organising, together with Siegfried Echterhoff, a series of lectures that are meant to provide basic knowledge of KK-theory. During the first three weeks of the semester, Siegfried Echterhoff will talk about crossed products of C*-algebras and locally compact groups, whereas I'm going to give an introduction to KK-theory in the Kasparov picture. There will be additional mini-courses by us in the advanced section of the semester, and I'm going to update this site as soon as I know more details.

Introduction to KK-theory

Schedule

The lectures take place twice a week, 90 minutes each.

Week	Topics
4.5.-8.5.	(Graded) C*-algebras, Hilbert modules, Kasparov's stabilisation theorem, definition of KK-theory, basic properties
11.5.-15.5.	Connections, Kasparov's Techincal Theorem, the Kasparov product
18.5.-22.5.	The Kasparov product (continued), Bott periodicity

Necessary preliminaries

The following topics seem necessary for the understanding of the course but will only covered shortly in the lectures; it is hence advisable that you make yourself acquainted with them before the lectures start:

• C*-algebra basics (definition, *-homomorphisms, ideals, quotients, approximate units, examples);
• K-theory for (locally) compact spaces, Banach algebras or C*-algebras (definition and at least superficial knowledge of basic properties like Bott periodicity and the six-term exact sequence);
• Fredholm operators on Hilbert spaces (definition, index).

An incomplete list of bibliographic references

Here is a pdf-file with some bibliographic references (mainly to text books); I hope that this is helpful for your preparation.

Lecture notes

• LaTeXed lecture notes (typed by Lin Shan)
• Hilbert modules, adjoinable and compact operators (handwritten)
• Kasparov's stabilisation theorem, tensor products, graded algebras (handwritten)
• Definition of KK-theory, basic properties such as bifunctoriality (handwritten)
• The standard simplifications, the Fredholm picture, KK(\C,B) = K_0(B) (three sketches) (handwritten)
• The Kasparov product: connections, existence (handwritten)
• The Kasparov product: existence continued, a useful lemma (handwritten)
• The Kasparov product: existence (LaTeXed notes for a seminar talk (2004), revised version)
• Overview: The general form of the Kasparov product, long exact sequences
• Bott peridocity of KK-theory (LaTeXed version)

Note that the partitioning of the lecture notes does not correspond to individual lectures but rather to natural chapters. Lin Shan has typed most of the lecture notes, so (most of) the handwritten notes are redundant. Thank you, Lin!