In 2022 spring semester, we ran a reading seminar on K-theory and KK-theory at Leiden university. In this seminar we studied some basic knowledge of KK-theory for C*-algebras. We began with a quick recap of the properties of K-theory, focusing on its formal properties like the Bott periodicity and Thom isomorphisms, which became more illuminating in the framework of KK-theory. Then we provided the two basic pictures of KK-theory due to Kasparov and Cuntz. After that we studied the formal properties of KK-theory, which are mostly due to the existence of the Kasparov product. Using the Kasparov product we are able to construct a category KK which is equipped with a triangulated structure, so various toolkits from homological algebra can be adapted to it. In the end, we discussed the unbounded picture of KK-theory, whose theory had been developed rapidly in recent years and proved to be useful in the study of noncommutative differential geometry.
Prerequisites. The participants should be familiar with basic theory of C*-algebras. Knowledge on the K-theory of C*-algebras is not assumed but suggested.
Bram Mesland, Jack Ekenstam, Yufan Ge and Yuezhao Li. If you are interested: please contact Yuezhao Li (y.li AT math.leidenuniv.nl).
Tuesdays, 12.15-14.00, in Gorlaeus building.
Notes of the talks are available here.
|K-theory of C*-algebras 1
|K-theory of C*-algebras 2
|Hilbert C*-modules 1
|Jack Ekenstam, Yufan Ge
|Hilbert C*-modules 2, Kasparov’s picture of KK-theory 1
|Kasparov’s picture of KK-theory 2
|Cuntz’s picture of KK-theory
|Examples and properties of KK-theory
|The Kasparov product 1
|The Kasparov product 2
|Extension of C*-algebras and KK-theory 1
|Georg Huppertz, Yuezhao Li
|Extension of C*-algebras and KK-theory 2, Categorical aspect of KK-theory 1
|Categorical aspect of KK-theory 2
|Finite summability in K-homology
|No talk (due to Noncommutative Geometry Along the North Sea 2022)
|K-theory of graph C*-algebras
|K-theory of Cuntz-Pimsner algebras