An introduction to $C^*$-algebras and $K$-theory

This course offers an introductory exploration of the fundamental principles of $C^*$-algebras and $K$-theory with emphasis on classification of $C^*$-algebras using K-theoretic data. The first part of this course delves into the essential aspects of $C^*$-algebras, which are helpful to introducing $K$-theory. These include the Gelfand representation, Gelfand-Naimark-Segal construction, and Gelfand-Naimark representation. The second part is devoted to the introduction of $K_0$-groups for $C^*$-algebras, covering their basic properties and exploring Elliott's classification of AF-algebras. Finally, the course proceeds to introduce $K_1$-groups. (For those who have taken the course with the same title in the autumn semester of 2023, the essential part of this course remains the same, while you can find discussions of new topics and adapted expositions of the familiar parts to new discourses along the process.)

Lecturer

Date

9th September ~ 5th December, 2024

Location

Weekday | Time | Venue | Online | ID | Password |
---|---|---|---|---|---|

Monday,Wednesday | 14:20 - 16:05 | A7-201 | ZOOM 2 | 638 227 8222 | BIMSA |

Prerequisite

Undergraduate Functional Analysis, General Topology, Algebra

Syllabus

1. Fundamentals of the theory of $C^*$-algebras

2. Projections and unitaries

3. $K_0$-groups for unital $C^*$-algebras

4. $K_0$-groups for general $C^*$-algebras

5. Order structure of $K_0$-groups

6. Inductive limit $C^*$-algebras

7. Classification of AF-algebras

8. $K_1$-groups for $C^*$-algebras

2. Projections and unitaries

3. $K_0$-groups for unital $C^*$-algebras

4. $K_0$-groups for general $C^*$-algebras

5. Order structure of $K_0$-groups

6. Inductive limit $C^*$-algebras

7. Classification of AF-algebras

8. $K_1$-groups for $C^*$-algebras

Reference

1. Gerard Murphy, $C^*$-algebras and operator theory, Academic Press, Inc., Boston, MA, 1990. MR1074574 (91m:46084)

2. Mikael Rørdam, Flemming Larsen, and Niels Laustsen, An introduction to K-theory for $C^*$-algebras, London Mathematical Society Student Texts, vol. 49, Cambridge University Press, Cambridge, 2000. MR1783408 (2001g:46001)

3. Karen Strung, An introduction to $C^*$-algebras and the classification program, Advanced Course in Mathematics, Birkhaüser, CRM, Barcelona, 2021. MR 4225279

4. Ronald Douglas, Banach algebra techniques in operator theory. Second edition. Graduate Texts in Mathematics, 179. Springer-Verlag, New York, 1998. MR1634900 (99c:47001)

5. John Conway, A course in operator theory. Graduate Studies in Mathematics, 21. American Mathematical Society, Providence, RI, 2000. MR1721402 (2001d:47001)

6. Bruce Blackadar, $K$-theory for operator algebras. Second edition. Mathematical Sciences Research Institute Publications, 5. Cambridge University Press, Cambridge, 1998. MR1656031 (99g:46104)

2. Mikael Rørdam, Flemming Larsen, and Niels Laustsen, An introduction to K-theory for $C^*$-algebras, London Mathematical Society Student Texts, vol. 49, Cambridge University Press, Cambridge, 2000. MR1783408 (2001g:46001)

3. Karen Strung, An introduction to $C^*$-algebras and the classification program, Advanced Course in Mathematics, Birkhaüser, CRM, Barcelona, 2021. MR 4225279

4. Ronald Douglas, Banach algebra techniques in operator theory. Second edition. Graduate Texts in Mathematics, 179. Springer-Verlag, New York, 1998. MR1634900 (99c:47001)

5. John Conway, A course in operator theory. Graduate Studies in Mathematics, 21. American Mathematical Society, Providence, RI, 2000. MR1721402 (2001d:47001)

6. Bruce Blackadar, $K$-theory for operator algebras. Second edition. Mathematical Sciences Research Institute Publications, 5. Cambridge University Press, Cambridge, 1998. MR1656031 (99g:46104)

Audience

Advanced Undergraduate
, Graduate
, Postdoc

Video Public

No

Notes Public

No

Language

English

Lecturer Intro

He graduated from the University of Tokyo in 2018, and then became an associate research fellow at University of Tokyo. After finishing his postdocteral position at East China Normal University in 2022, He joined BIMSA as an assistant professor in 2023. He's recent research interests lie in classification theory of C*-algebras, C*-dynamical systems and topological dynamical systems.