Prof. Dr. Moritz Weber

K-Theory of C*-Algebras

(Winter term 2019/2020)


Please contact Moritz Weber regarding the oral exams.

Time and Place

Lecture (14 Oct - 7 Dec):
Tuesday, 10:15-11:45, SR2
Thursday, 16:00-17:30, SR10

Exercise session (28 Oct - 7 Dec):
Monday, 10:15-11:45, HS IV

Lecture notes

tex form (in English eventually), written by Friedrich:
K-Theory for C*-algebras

scanned lecture notes (in German), hand-written by Moritz:
Chapter 1 Equivalence relations on unitaries
Chapter 2 Equivalence relations on projections
Chapter 3 Definition of K0
Chapter 4 Functoriality, additivity, finite stability, homotopy invariance and the standard picture of K0
Chapter 5 Continuity and stability of K0
Chapter 6 Half exactness and split exactness of K0
Chapter 7 K1
Chapter 8 Long exact sequences
Chapter 9 Excurs: nuclearity and exactness
Chapter 10 Bott periodicity

Exercise sheets:
Sheet 1 K-theory for commutative C*-algebras
(28 Oct: 1 - Friedrich, 2 - Luca, 3 - Mirko)
(11 Nov: 4 - Steven, 5 - Moritz Sp.)
Sheet 2 K-theory for AF algebras
(18 Nov: 1 - Marcel, 2 - Julien)
(25 Nov: 3 - Mirko, 4 - ?)
(2 Dec: 5 - ?, 6 - ?)


In this lecture, we will introduce K-theory for C*-algebras. This is a theory of invariants for C*-algebras
with a homological flavour. More concretely, to any C*-algebra A we assign an abelian group K0(A)
which somehow "counts the projections", as well as an abelian group K1(A) which somehow "count
s the unitaries". Almost more important than the definition of the K-groups are the homological
properties of the K functor: it preserves many natural constructions making it much simpler to
compute the K-groups in concrete cases.

See also summer term 2012 for a previous variant of this lecture.

Participants should know the definition and some basics on C*-algebras as well as functional analysis.


Rordam, Mikael; Larsen, Flemming; Laustsen, Niels, An introduction to K-theory for C*-algebras, 2000.
Blackadar, Bruce, K-theory for operator algebras, 1998.
Wegge-Olsen, Niels, K-theory and C*-algebras. A friendly approach, 1993.
Blackadar, Bruce, Operator algebras. Theory of C*-algebras and von Neumann algebras, 2006.
Brown, Nathanial; Ozawa, Narutaka, C*-algebras and finite-dimensional approximations, 2008.
Davidson, Kenneth, C*-algebras by example, 1996.
Lecture notes by Christian Voigt

some further literature

Last update: 15 May 2020   Moritz Weber Impressum