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.)
讲师
日期
2024年09月09日 至 12月05日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周一,周三 | 14:20 - 16:05 | A7-201 | ZOOM 2 | 638 227 8222 | BIMSA |
修课要求
Undergraduate Functional Analysis, General Topology, Algebra
课程大纲
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
参考资料
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)
听众
Advanced Undergraduate
, Graduate
, 博士后
视频公开
不公开
笔记公开
不公开
语言
英文
讲师介绍
贺卓丰2018年毕业于东京大学，后成为该校副研究员。2022年华东师范大学博士后出站，2023年加入北京雁栖湖应用数学研究院任助理研究员。现在的研究兴趣包括C*代数的分类理论、C*动力系统和拓扑动力系统。