K-theory for C*-algebras, and beyond

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.

Reference

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.

Keywords

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

Required Knowledge

Knowledge on standard undergraduate functional analysis.

Attendance

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

• Introduction

full document (110 pages)

• K-theory for C*-algebras, and beyond

Lecture Notes

Title

Contents

Chapter 1

• C*-algebras

Chapter 2

• Projections and unitary elements

Chapter 3

• K_0-group for a unital C*-algebra

Chapter 4

• K_0-group for an arbitrary C*-algebra

Chapter 5

• The functor K_1

Chapter 6

• The index map

Chapter 7

• Higher K-functors, Bott periodicity

Chapter 8

• The six-term exact sequence

Chapter 9

• Cyclic cohomology

Chapter 10

• Application: Levinson's theorem

Bibliography

• Bibliography

Method of Evaluation

Grades based on attendance, written reports, and discussions.

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