K-theory for C*-algebras, and beyond

Serge RICHARD Professor

Department: Graduate School of Mathematics

Class Time: 2015 Spring Friday
Recommended for: Graduate school of Mathematics

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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.

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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.

Method of Evaluation

Grades based on attendance, written reports, and discussions.

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Lecture Notes

First lecture
Introduction (PDF, 40KB)
full document (110 pages)
K-theory for C*-algebras, and beyond (PDF, 453KB)

Lecture Notes

Title (PDF, 33KB)
Contents (PDF, 58KB)
Chapter 1
C*-algebras (PDF, 132KB)
Chapter 2
Projections and unitary elements (PDF, 135KB)
Chapter 3
K_0-group for a unital C*-algebra (PDF, 132KB)
Chapter 4
K_0-group for an arbitrary C*-algebra (PDF, 110KB)
Chapter 5
The functor K_1 (PDF, 108KB)
Chapter 6
The index map (PDF, 112KB)
Chapter 7
Higher K-functors, Bott periodicity (PDF, 129KB)
Chapter 8
The six-term exact sequence (PDF, 93KB)
Chapter 9
Cyclic cohomology (PDF, 193KB)
Chapter 10
Application: Levinson's theorem (PDF, 128KB)
Bibliography (PDF, 51KB)

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Page last updated August 7, 2015

The class contents were most recently updated on the date indicated. Please be aware that there may be some changes between the most recent year and the current page.

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