### Introduction to equivariant KK-theory and the Baum-Connes conjecture

This is the webpage of a project course on KK-theory, run in Block 2 2013 at the University of Copenhagen. If you have any questions about the course, please [contact me].

#### Schedule:

We meet each Tuesday afternoon, 13:30 - 15:30, in room 04.4.01 (seminar room on the fourth floor of the math department).

#### Projects:

To get credit for the course, you need to write a project on some topic related to KK-theory. Please talk to me about possible topics.

#### Weekly plan:

(tentative)

Week 1 [19 nov] Speaker: Tyrone. Topic: K-homology.
Reading: Higson-Roe, Section 2.3; Section 8.1 (skip 8.1.5, 8.1.7, 8.1.11 and everything else about multigradings, unless you're particularly interested in them); Section 8.2 (skip 8.2.13); Section 8.7 (just read 8.7.1, 8.7.2 and 8.7.4).
Problems: Higson-Roe, Exercises 2.9.3, 8.8.1, 8.8.11, 8.8.12 and 8.8.13.
If you have extra time, also read section 8.3 and try problems 8.8.15 and 8.8.16.

Week 2 [26 nov] Speaker: Niek de Kleijn. Topic: Hilbert modules.
Reading: Lance: intro to chapter 1, pages 1-2. Blackadar: 13.1, 13.2, 12.1, 12.2, 12.3, 13.4, 13.5, 13.6.
Problems: Fill in the missing details from the reading (possibly with the help of Jensen-Thomsen or Lance). Jensen-Thomsen: exercises E1.1.1, 1.1.3, 1.1.6. Also, give an example of a non-adjointable linear map between Hilbert modules.
More reading: If you haven't encountered graded algebras/modules before, look through Blackadar's chapter 14 in preparation for the following weeks.

Week 3 [3 dec] Speaker: Kang Li. Topic: The KK-groups.
Problems: Jensen-Thomsen, E2.1.1 and 2.1.4. And fill in the missing details from the reading.

Week 4 [10 dec] No meeting this week. Topic: The Kasparov product I.
Reading: Blackadar: Theorems 12.4.2 and 14.6.2 ('Kasparov's Technical Theorem'); 18.1, 18.2, 18.3, 18.4.

Week 5 [17 dec] Speaker: Andrew Leitch. Topic: The Kasparov product II.
Reading: Blackadar: 18.5, 18.6, 18.7, 18.8, 18.9, 18.12. If you still have time and energy: 18.10.

Meetings after the Christmas / New Year holiday will be devoted to project presentations and discussion of some more specialised topics.

Week 6 [7 jan] Speaker: Dominic Enders. Topic: Cuntz's picture of KK-theory

Week 7 [14 jan] Speaker: Ryszard Nest. Topic: The KK-category.

Week 8 [21 jan] Speaker: Niek de Kleijn. Topic: On the UCT in KK-theory

Week 9 [29 jan] Speakers: Andrew Leitch (Topic: The Noncommutative Atiyah-Jänich Theorem) and Clarisson Rizzie Canlubo (Topic: The Baum-Connes Conjecture)

Main references:

• Blackadar: K-theory for operator algebras. Second edition. Cambridge University Press, 1998.
• Jensen and Thomsen: Elements of KK-theory. Birkhäuser, 1991.
• Higson and Roe: Analytic K-homology . Oxford University Press, 2000.

Other sources: for background, examples, possible projects, etc., listed in no particular order. The list will grow as we go along, but will never be exhaustive.

• Kasparov: Operator K-theory and its applications: elliptic operators, group representations, higher signatures, C*-extensions. Proceedings of the ICM, 1983. [pdf]
• Kasparov: K-theory, group C*-algebras, and higher signatures (conspectus). London Math. Soc. Lecture Note Ser., 226, Cambridge Univ. Press, Cambridge, 1995. pp. 101-146.
• Higson: The Baum-Connes conjecture. Proceedings of the ICM, 1998. [pdf]
• Higson: A primer on KK-theory. Proc. Sympos. Pure Math., 51, Part 1, Amer. Math. Soc., Providence, RI, 1990. pp. 239-283 [pdf]
• Cuntz: K-theory and C*-algebras. Algebraic K-theory, number theory, geometry and analysis (Bielefeld, 1982), 55–79, Lecture Notes in Math., 1046, Springer, Berlin, 1984.
• Atiyah: Global theory of elliptic operators. Proc. Internat. Conf. on Functional Analysis and Related Topics (Tokyo, 1969) pp. 21–30
• Atiyah: Algebraic topology and operators in Hilbert space. 1969 Lectures in Modern Analysis and Applications. I pp. 101–121 Springer.
• Baum, Connes and Higson: Classifying space for proper actions and K-theory of group C*-algebras. C*-algebras: 1943–1993 (San Antonio, TX, 1993), 240–291, Contemp. Math., 167, AMS 1994. [pdf]
• Brown, Douglas and Fillmore: Extensions of C*-algebras and K-homology. Ann. of Math. (2) 105 (1977), no. 2, 265–324.
• Valette: Introduction to the Baum-Connes conjecture. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2002. [pdf]
• Connes: Noncommutative geometry. Academic Press, 1994. [pdf]
• Lance: Hilbert C*-modules: a toolkit for operator algebraists. London Math. Soc. Lecture Note Series 210. Cambridge Univ. Press, Cambridge, 1995.