K-theory for C*-algebras, and beyond

A dinosaur
LecturerSerge RICHARD, Professor
DepartmentGraduate School of Mathematics, 2015 Spring
Recommended for:Graduate school of Mathematics (21.5 hours / session One session / week 15 weeks / semester)

Course Overview

This course will provide an overview of some recent tools introduced at the crossroad between functional analysis, geometry and operator algebras. It can be considered as a course on non-commutative topology, which is the first step toward non-commutative geometry. In order to provide a large panorama on the subject together with applications, some details might be omitted, but references for all proofs will be provided.


There is no specific book related to this course. References and additional material will be provided during the lectures.

Plan of the course

Tentative program:

  1. C*-algebras,
  2. Projections and unitaries,
  3. K0 and its properties,
  4. K1 and its properties,
  5. Index map and Bott periodicity,
  6. The six-term exact sequence,
  7. Cyclic cohomology,
  8. Connes' pairing,
  9. Applications.


C*-algebras, K-theory, index map, cyclic cohomology.

Required Knowledge

Knowledge on standard undergraduate functional analysis.


This course is open for any students at Nagoya University as one of the “open subjects” of general education.

Additional advice

Lecture notes will be provided for this course.

Lecture Notes

First lecture

full document (110 pages)

Lecture Notes



Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10


Method of Evaluation

Grades based on attendance, written reports, and discussions.

Last updated

December 23, 2019