## Prof. Dr. Moritz Weber

## K-Theory of C*-Algebras

(Winter term 2019/2020)### News

**The K-Theory lecture is over!**(no session on Thu, 5 Dec)

From 10 December, Michael Hartz will give an (independent) lecture on Hardy spaces at the same

time and place as the K-theory lecture.

*Oral exams:*

Mon, 16 Dec, 11:30 Mirko, 12:00 Marcel, 12:30 Luca

### 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:*

Introduction

Chapter 1 Equivalence relations on unitaries

Chapter 2 Equivalence relations on projections

Chapter 3 Definition of K

_{0}

Chapter 4 Functoriality, additivity, finite stability, homotopy invariance and the standard picture of K

_{0}

Chapter 5 Continuity and stability of K

_{0}

Chapter 6 Half exactness and split exactness of K

_{0}

Chapter 7 K

_{1}

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 - ?)

### Contents

In this lecture, we will introduce K-theory for C*-algebras. This is a theory of invariants for C*-algebraswith a homological flavour. More concretely, to any C*-algebra

*A*we assign an abelian group

*K*

_{0}(A)which somehow "counts the projections", as well as an abelian group

*K*which somehow "count

_{1}(A)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.

### Literature

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: 21 November 2019 Moritz Weber | Impressum |